dataset
stringclasses 1
value | solutions
null | task_id
int64 0
467
| exe_method
stringclasses 1
value | example_input
sequencelengths 1
5
| example_output
sequencelengths 1
5
| test_input
sequencelengths 1
20
| test_output
sequencelengths 1
20
| question
stringlengths 474
5.27k
| test_time_limit
int64 1
1
|
|---|---|---|---|---|---|---|---|---|---|
Codeforces_test | null | 401 | stdin | [
"8\n3 9 2\n4 9 1\n7 9 2\n2 10 2\n154 220 2\n147 294 2\n998 24435 3\n1 1000000000 2\n"
] | [
"2\n6\n0\n4\n0\n1\n7148\n500000000\n"
] | [
"8\n3 9 2\n4 9 1\n7 9 2\n2 10 2\n154 220 2\n147 294 2\n998 24435 3\n1 1000000000 2\n"
] | [
"2\n6\n0\n4\n0\n1\n7148\n500000000\n"
] | You are given a positive integer $$$k$$$ and a set $$$S$$$ of all integers from $$$l$$$ to $$$r$$$ (inclusive).
You can perform the following two-step operation any number of times (possibly zero):
1. First, choose a number $$$x$$$ from the set $$$S$$$, such that there are at least $$$k$$$ multiples of $$$x$$$ in $$$S$$$ (including $$$x$$$ itself);
2. Then, remove $$$x$$$ from $$$S$$$ (note that nothing else is removed).
Find the maximum possible number of operations that can be performed.
Input format:
Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1\le t\le 10^4$$$) — the number of test cases. The description of test cases follows.
The only line of each test case contains three integers $$$l$$$, $$$r$$$, and $$$k$$$ ($$$1\le l\le r\leq 10^9$$$, $$$1\leq k\le r-l+1$$$) — the minimum integer in $$$S$$$, the maximum integer in $$$S$$$, and the parameter $$$k$$$.
Output format:
For each test case, output a single integer — the maximum possible number of operations that can be performed.
Example Input 0:
8
3 9 2
4 9 1
7 9 2
2 10 2
154 220 2
147 294 2
998 24435 3
1 1000000000 2
Example Output 0:
2
6
0
4
0
1
7148
500000000
Notes:
In the first test case, initially, $$$S = \{3,4,5,6,7,8,9\}$$$. One possible optimal sequence of operations is:
1. Choose $$$x = 4$$$ for the first operation, since there are two multiples of $$$4$$$ in $$$S$$$: $$$4$$$ and $$$8$$$. $$$S$$$ becomes equal to $$$\{3,5,6,7,8,9\}$$$;
2. Choose $$$x = 3$$$ for the second operation, since there are three multiples of $$$3$$$ in $$$S$$$: $$$3$$$, $$$6$$$, and $$$9$$$. $$$S$$$ becomes equal to $$$\{5,6,7,8,9\}$$$.
In the second test case, initially, $$$S=\{4,5,6,7,8,9\}$$$. One possible optimal sequence of operations is:
1. Choose $$$x = 5$$$, $$$S$$$ becomes equal to $$$\{4,6,7,8,9\}$$$;
2. Choose $$$x = 6$$$, $$$S$$$ becomes equal to $$$\{4,7,8,9\}$$$;
3. Choose $$$x = 4$$$, $$$S$$$ becomes equal to $$$\{7,8,9\}$$$;
4. Choose $$$x = 8$$$, $$$S$$$ becomes equal to $$$\{7,9\}$$$;
5. Choose $$$x = 7$$$, $$$S$$$ becomes equal to $$$\{9\}$$$;
6. Choose $$$x = 9$$$, $$$S$$$ becomes equal to $$$\{\}$$$.
In the third test case, initially, $$$S=\{7,8,9\}$$$. For each $$$x$$$ in $$$S$$$, no multiple of $$$x$$$ other than $$$x$$$ itself can be found in $$$S$$$. Since $$$k = 2$$$, you can perform no operations.
In the fourth test case, initially, $$$S=\{2,3,4,5,6,7,8,9,10\}$$$. One possible optimal sequence of operations is:
1. Choose $$$x = 2$$$, $$$S$$$ becomes equal to $$$\{3,4,5,6,7,8,9,10\}$$$;
2. Choose $$$x = 4$$$, $$$S$$$ becomes equal to $$$\{3,5,6,7,8,9,10\}$$$;
3. Choose $$$x = 3$$$, $$$S$$$ becomes equal to $$$\{5,6,7,8,9,10\}$$$;
4. Choose $$$x = 5$$$, $$$S$$$ becomes equal to $$$\{6,7,8,9,10\}$$$. | 1 |
Codeforces_test | null | 402 | stdin | [
"3\n6 1 1\n1 3 3\n10 1 2\n"
] | [
"14\n1\n53\n"
] | [
"3\n6 1 1\n1 3 3\n10 1 2\n",
"1\n1 1 1\n",
"1\n437300461 96625 34833\n",
"1\n910351181 78173 15387\n",
"1\n793467309 94313 52836\n",
"1\n971550733 43158 33390\n",
"1\n854666861 92002 38136\n",
"10\n33854166 29066 49060\n32759898 53835 75905\n78083558 64545 62296\n47722624 64272 48425\n3707310 69455 39398\n54203928 38467 3960\n31571982 10075 68047\n65568487 56019 35649\n66022256 60400 12229\n46632160 43622 10345\n",
"10\n11937590 77910 62317\n63422079 46192 21977\n12888245 48533 96451\n24994103 24712 75464\n41507401 63185 42425\n31651845 43085 84222\n10035408 50123 55987\n85716265 63529 10033\n93368817 70608 6450\n32089725 90638 86023\n",
"10\n95053718 26754 67063\n89051557 28150 24946\n87884420 32521 63311\n7298285 85151 35208\n79307491 65426 80044\n14132466 90808 31780\n78433426 47068 52440\n60639851 3742 75904\n20715378 80817 67968\n57738778 72247 85893\n",
"10\n68104438 42895 47617\n19713738 53211 3723\n12623700 16509 97466\n94635171 78295 70760\n71883389 59155 74559\n91580383 38530 36234\n51864148 11309 16189\n80787628 87059 50287\n43029235 23729 62189\n38163639 62367 20356\n",
"10\n51220566 24443 85066\n90567408 2465 49796\n47428387 497 31622\n76939354 38735 30503\n14716183 61396 44882\n14252492 43149 83792\n20262166 84062 4129\n935406 61864 24670\n65343092 9745 56410\n18588500 43975 20226\n",
"1\n665120731 50406 87091\n",
"1\n843204155 99251 24540\n",
"1\n21287579 48095 5094\n",
"1\n199371003 72747 85648\n",
"1\n82487131 21592 23097\n",
"1\n260570555 37732 36355\n",
"1\n438653979 86577 41101\n",
"1\n1000000000 100000 100000\n"
] | [
"14\n1\n53\n",
"1\n",
"49488359\n",
"400820889\n",
"782606669\n",
"69892732\n",
"228818698\n",
"709113841\n853853654\n191196773\n702907510\n397887101\n320084139\n818836875\n661913564\n916869136\n216310942\n",
"725044695\n564214506\n974277617\n939267073\n206169492\n792733781\n432236563\n484626689\n916449219\n657450118\n",
"695688191\n770056423\n606377588\n872214590\n843944293\n626155071\n707563713\n692599575\n190797559\n305057518\n",
"147112248\n912938221\n132963514\n80861124\n242654962\n771240444\n625522388\n713215583\n357663073\n944127626\n",
"739004279\n39067574\n124576989\n396021404\n357521528\n838157718\n233645370\n372466701\n757096460\n988887997\n",
"566793373\n",
"604828018\n",
"732697221\n",
"150837339\n",
"715887445\n",
"283581242\n",
"576188257\n",
"388462163\n"
] | Let $$$n$$$ and $$$d$$$ be positive integers. We build the the divisor tree $$$T_{n,d}$$$ as follows:
- The root of the tree is a node marked with number $$$n$$$. This is the $$$0$$$-th layer of the tree.
- For each $$$i$$$ from $$$0$$$ to $$$d - 1$$$, for each vertex of the $$$i$$$-th layer, do the following. If the current vertex is marked with $$$x$$$, create its children and mark them with all possible distinct divisors$$$^\dagger$$$ of $$$x$$$. These children will be in the $$$(i+1)$$$-st layer.
- The vertices on the $$$d$$$-th layer are the leaves of the tree.
For example, $$$T_{6,2}$$$ (the divisor tree for $$$n = 6$$$ and $$$d = 2$$$) looks like this:
Define $$$f(n,d)$$$ as the number of leaves in $$$T_{n,d}$$$.
Given integers $$$n$$$, $$$k$$$, and $$$d$$$, please compute $$$\sum\limits_{i=1}^{n} f(i^k,d)$$$, modulo $$$10^9+7$$$.
$$$^\dagger$$$ In this problem, we say that an integer $$$y$$$ is a divisor of $$$x$$$ if $$$y \ge 1$$$ and there exists an integer $$$z$$$ such that $$$x = y \cdot z$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The only line of each test case contains three integers $$$n$$$, $$$k$$$, and $$$d$$$ ($$$1 \le n \le 10^9$$$, $$$1 \le k,d \le 10^5$$$).
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^9$$$.
Output format:
For each test case, output $$$\sum\limits_{i=1}^{n} f(i^k,d)$$$, modulo $$$10^9+7$$$.
Example Input 0:
3
6 1 1
1 3 3
10 1 2
Example Output 0:
14
1
53
Notes:
In the first test case, $$$n = 6$$$, $$$k = 1$$$, and $$$d = 1$$$. Thus, we need to find the total number of leaves in the divisor trees $$$T_{1,1}$$$, $$$T_{2,1}$$$, $$$T_{3,1}$$$, $$$T_{4,1}$$$, $$$T_{5,1}$$$, $$$T_{6,1}$$$.
- $$$T_{1,1}$$$ has only one leaf, which is marked with $$$1$$$.
- $$$T_{2,1}$$$ has two leaves, marked with $$$1$$$ and $$$2$$$.
- $$$T_{3,1}$$$ has two leaves, marked with $$$1$$$ and $$$3$$$.
- $$$T_{4,1}$$$ has three leaves, marked with $$$1$$$, $$$2$$$, and $$$4$$$.
- $$$T_{5,1}$$$ has two leaves, marked with $$$1$$$ and $$$5$$$.
- $$$T_{6,1}$$$ has four leaves, marked with $$$1$$$, $$$2$$$, $$$3$$$, and $$$6$$$.
The total number of leaves is $$$1 + 2 + 2 + 3 + 2 + 4 = 14$$$.
In the second test case, $$$n = 1$$$, $$$k = 3$$$, $$$d = 3$$$. Thus, we need to find the number of leaves in $$$T_{1,3}$$$, because $$$1^3 = 1$$$. This tree has only one leaf, so the answer is $$$1$$$. | 1 |
Codeforces_test | null | 403 | stdin | [
"3\n4 2 2\n1 2 3 4\n1 1\n1 2\n1 1\n3 6 2\n1 2 3\n1 1 2 3 3 2\n3 3\n2 2\n4 6 2\n3 1 4 2\n3 1 1 2 3 4\n3 4\n4 2\n"
] | [
"YA\nTIDAK\nYA\nYA\nTIDAK\nYA\nTIDAK\nYA\nYA\n"
] | [
"3\n4 2 2\n1 2 3 4\n1 1\n1 2\n1 1\n3 6 2\n1 2 3\n1 1 2 3 3 2\n3 3\n2 2\n4 6 2\n3 1 4 2\n3 1 1 2 3 4\n3 4\n4 2\n"
] | [
"YA\nTIDAK\nYA\nYA\nTIDAK\nYA\nTIDAK\nYA\nYA\n"
] | This is the hard version of the problem. In the two versions, the constraints on $$$q$$$ and the time limit are different. In this version, $$$0 \leq q \leq 2 \cdot 10^5$$$. You can make hacks only if all the versions of the problem are solved.
A team consisting of $$$n$$$ members, numbered from $$$1$$$ to $$$n$$$, is set to present a slide show at a large meeting. The slide show contains $$$m$$$ slides.
There is an array $$$a$$$ of length $$$n$$$. Initially, the members are standing in a line in the order of $$$a_1, a_2, \ldots, a_n$$$ from front to back. The slide show will be presented in order from slide $$$1$$$ to slide $$$m$$$. Each section will be presented by the member at the front of the line. After each slide is presented, you can move the member at the front of the line to any position in the lineup (without changing the order of the rest of the members). For example, suppose the line of members is $$$[\color{red}{3},1,2,4]$$$. After member $$$3$$$ presents the current slide, you can change the line of members into either $$$[\color{red}{3},1,2,4]$$$, $$$[1,\color{red}{3},2,4]$$$, $$$[1,2,\color{red}{3},4]$$$ or $$$[1,2,4,\color{red}{3}]$$$.
There is also an array $$$b$$$ of length $$$m$$$. The slide show is considered good if it is possible to make member $$$b_i$$$ present slide $$$i$$$ for all $$$i$$$ from $$$1$$$ to $$$m$$$ under these constraints.
However, your annoying boss wants to make $$$q$$$ updates to the array $$$b$$$. In the $$$i$$$-th update, he will choose a slide $$$s_i$$$ and a member $$$t_i$$$ and set $$$b_{s_i} := t_i$$$. Note that these updates are persistent, that is changes made to the array $$$b$$$ will apply when processing future updates.
For each of the $$$q+1$$$ states of array $$$b$$$, the initial state and after each of the $$$q$$$ updates, determine if the slideshow is good.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains three integers $$$n$$$, $$$m$$$ and $$$q$$$ ($$$1 \le n, m \le 2 \cdot 10^5$$$; $$$0 \leq q \leq 2 \cdot 10^5$$$) — the number of members and the number of sections.
The second line of each test case contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \le a_i \le n$$$) — the initial order of the members from front to back. It is guaranteed that each integer from $$$1$$$ to $$$n$$$ appears exactly once in $$$a$$$.
The third line of each test case contains $$$m$$$ integers $$$b_1, b_2, \ldots, b_m$$$ ($$$1 \le b_i \le n$$$) — the members who should present each section.
Each of the next $$$q$$$ lines contains two integers $$$s_i$$$ and $$$t_i$$$ ($$$1 \le s_i \le m$$$, $$$1 \le t_i \le n$$$) — parameters of an update.
It is guaranteed that the sum of $$$n$$$, the sum of $$$m$$$ and the sum of $$$q$$$ over all test cases do not exceed $$$2 \cdot 10^5$$$ respectively.
Output format:
For each test case, output $$$q+1$$$ lines corresponding to the $$$q+1$$$ states of the array $$$b$$$. Output "YA" if the slide show is good, and "TIDAK" otherwise.
You can output the answer in any case (upper or lower). For example, the strings "yA", "Ya", "ya", and "YA" will be recognized as positive responses.
Example Input 0:
3
4 2 2
1 2 3 4
1 1
1 2
1 1
3 6 2
1 2 3
1 1 2 3 3 2
3 3
2 2
4 6 2
3 1 4 2
3 1 1 2 3 4
3 4
4 2
Example Output 0:
YA
TIDAK
YA
YA
TIDAK
YA
TIDAK
YA
YA
Notes:
For the first test case, you do not need to move the members as both slides are presented by member $$$1$$$, who is already at the front of the line. After that, set $$$b_1 := 2$$$, now slide $$$1$$$ must be presented by member $$$2$$$ which is impossible as member $$$1$$$ will present slide $$$1$$$ first. Then, set $$$b_1 = 1$$$, the $$$b$$$ is the same as the initial $$$b$$$, making a good presentation possible. | 1 |
Codeforces_test | null | 404 | stdin | [
"2\n6\n\n0\n\n2\n\n3\n\n5\n\n3\n"
] | [
"xor 1 1\n\nxor 2 2\n\nxor 3 3\n\nxor 4 6\n\nans 2 3 5\n\nans 1 2 3\n"
] | [
"2\n6 2 3 5\n3 1 2 3\n",
"7\n1000000000000000000 1000000000000000000 1 2\n1000000000000000000 2 1 1000000000000000000\n1000000000000000000 2 1000000000000000000 1\n1000000000000000000 4096 2048 6144\n1000000000000000000 864691128455135232 576460752303423488 288230376151711744\n864691128455135232 864691128455135232 576460752303423488 288230376151711744\n12288 8191 12287 12288\n",
"1\n19 17 18 19\n",
"1\n114514 114 514 1919\n"
] | [
"1\n1\n",
"1\n1\n1\n1\n1\n1\n1\n",
"1\n",
"1\n"
] | This is an interactive problem.
The Department of Supernatural Phenomena at the Oxenfurt Academy has opened the Library of Magic, which contains the works of the greatest sorcerers of Redania — $$$n$$$ ($$$3 \leq n \leq 10^{18}$$$) types of books, numbered from $$$1$$$ to $$$n$$$. Each book's type number is indicated on its spine. Moreover, each type of book is stored in the library in exactly two copies! And you have been appointed as the librarian.
One night, you wake up to a strange noise and see a creature leaving the building through a window. Three thick tomes of different colors were sticking out of the mysterious thief's backpack. Before you start searching for them, you decide to compute the numbers $$$a$$$, $$$b$$$, and $$$c$$$ written on the spines of these books. All three numbers are distinct.
So, you have an unordered set of tomes, which includes one tome with each of the pairwise distinct numbers $$$a$$$, $$$b$$$, and $$$c$$$, and two tomes for all numbers from $$$1$$$ to $$$n$$$, except for $$$a$$$, $$$b$$$, and $$$c$$$. You want to find these values $$$a$$$, $$$b$$$, and $$$c$$$.
Since you are not working in a simple library, but in the Library of Magic, you can only use one spell in the form of a query to check the presence of books in their place:
- "xor l r" — Bitwise XOR query with parameters $$$l$$$ and $$$r$$$. Let $$$k$$$ be the number of such tomes in the library whose numbers are greater than or equal to $$$l$$$ and less than or equal to $$$r$$$. You will receive the result of the computation $$$v_1 \oplus v_2 \oplus ... \oplus v_k$$$, where $$$v_1 ... v_k$$$ are the numbers on the spines of these tomes, and $$$\oplus$$$ denotes the operation of bitwise exclusive OR.
Since your magical abilities as a librarian are severely limited, you can make no more than $$$150$$$ queries.
Input format:
The first line of input contains an integer $$$t$$$ ($$$1 \le t \le 300$$$) — the number of test cases.
The first line of each test case contains a single integer $$$n$$$ ($$$3 \leq n \leq 10^{18}$$$) — the number of types of tomes.
Example Input 0:
2
6
0
2
3
5
3
Example Output 0:
xor 1 1
xor 2 2
xor 3 3
xor 4 6
ans 2 3 5
ans 1 2 3
Notes:
In the first test case, the books in the library after the theft look like this:
Now consider the answers to the queries:
- For the query "xor 1 1", you receive the result $$$1 \oplus 1 = 0$$$. Two tomes satisfy the condition specified in the query — both with the number $$$1$$$.
- For the query "xor 2 2", you receive the result $$$2$$$, as only one tome satisfies the specified condition.
- For the query "xor 3 3", you receive the result $$$3$$$.
- For the query "xor 4 6", you receive the result $$$4 \oplus 6 \oplus 4 \oplus 5 \oplus 6 = 5$$$.
In the second test case, there are only $$$3$$$ types of books, and it is easy to guess that the missing ones have the numbers $$$1$$$, $$$2$$$, and $$$3$$$. | 1 |
Codeforces_test | null | 405 | stdin | [
"5\n4\n4 2 1 3\n2\n1 2\n6\n5 4 1 3 2 6\n7\n5 6 1 3 7 2 4\n5\n3 1 2 5 4\n"
] | [
"1 1 2\n1\n1 1 2 1 3\n1 1 1 2 1 2\n1 1 2 1\n"
] | [
"5\n4\n4 2 1 3\n2\n1 2\n6\n5 4 1 3 2 6\n7\n5 6 1 3 7 2 4\n5\n3 1 2 5 4\n"
] | [
"1 1 2\n1\n1 1 2 1 3\n1 1 1 2 1 2\n1 1 2 1\n"
] | You are given a permutation $$$p$$$ of length $$$n$$$.
You can perform operations of two types:
- mark all positions $$$i$$$ such that $$$1 \le i < n$$$ and $$$p_i < p_{i + 1}$$$, and simultaneously remove the elements at these positions;
- mark all positions $$$i$$$ such that $$$2 \le i \le n$$$ and $$$p_{i - 1} > p_i$$$, and simultaneously remove the elements at these positions.
For each integer from $$$1$$$ to $$$(n-1)$$$, calculate the minimum number of operations required to remove that integer from the permutation.
Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.
The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 250\,000$$$).
The second line of each test case contains $$$n$$$ integers $$$p_1, p_2, \dots, p_n$$$ ($$$1 \le p_i \le n$$$). The array $$$p$$$ is a permutation.
Additional constraints on the input: the sum of $$$n$$$ over all test cases does not exceed $$$250\,000$$$.
Output format:
For each test case, print $$$(n-1)$$$ integers. The $$$i$$$-th of them should be equal to the minimum number of operations required to remove $$$i$$$ from the permutation.
Example Input 0:
5
4
4 2 1 3
2
1 2
6
5 4 1 3 2 6
7
5 6 1 3 7 2 4
5
3 1 2 5 4
Example Output 0:
1 1 2
1
1 1 2 1 3
1 1 1 2 1 2
1 1 2 1
| 1 |
Codeforces_test | null | 406 | stdin | [
"5\n1\n000\n1001\n10101\n01100101011101\n"
] | [
"1\n0\n2\n3\n8\n"
] | [
"5\n1\n000\n1001\n10101\n01100101011101\n",
"1\n00000000000000000000000000000000000000000000000000\n"
] | [
"1\n0\n2\n3\n8\n",
"0\n"
] | You are given a string $$$s$$$ of length $$$n$$$ consisting of $$$\mathtt{0}$$$ and/or $$$\mathtt{1}$$$. In one operation, you can select a non-empty subsequence $$$t$$$ from $$$s$$$ such that any two adjacent characters in $$$t$$$ are different. Then, you flip each character of $$$t$$$ ($$$\mathtt{0}$$$ becomes $$$\mathtt{1}$$$ and $$$\mathtt{1}$$$ becomes $$$\mathtt{0}$$$). For example, if $$$s=\mathtt{\underline{0}0\underline{101}}$$$ and $$$t=s_1s_3s_4s_5=\mathtt{0101}$$$, after the operation, $$$s$$$ becomes $$$\mathtt{\underline{1}0\underline{010}}$$$.
Calculate the minimum number of operations required to change all characters in $$$s$$$ to $$$\mathtt{0}$$$.
Recall that for a string $$$s = s_1s_2\ldots s_n$$$, any string $$$t=s_{i_1}s_{i_2}\ldots s_{i_k}$$$ ($$$k\ge 1$$$) where $$$1\leq i_1 < i_2 < \ldots <i_k\leq n$$$ is a subsequence of $$$s$$$.
Input format:
The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of input test cases.
The only line of each test case contains the string $$$s$$$ ($$$1\le |s|\le 50$$$), where $$$|s|$$$ represents the length of $$$s$$$.
Output format:
For each test case, output the minimum number of operations required to change all characters in $$$s$$$ to $$$\mathtt{0}$$$.
Example Input 0:
5
1
000
1001
10101
01100101011101
Example Output 0:
1
0
2
3
8
Notes:
In the first test case, you can flip $$$s_1$$$. Then $$$s$$$ becomes $$$\mathtt{0}$$$, so the answer is $$$1$$$.
In the fourth test case, you can perform the following three operations in order:
1. Flip $$$s_1s_2s_3s_4s_5$$$. Then $$$s$$$ becomes $$$\mathtt{\underline{01010}}$$$.
2. Flip $$$s_2s_3s_4$$$. Then $$$s$$$ becomes $$$\mathtt{0\underline{010}0}$$$.
3. Flip $$$s_3$$$. Then $$$s$$$ becomes $$$\mathtt{00\underline{0}00}$$$.
It can be shown that you can not change all characters in $$$s$$$ to $$$\mathtt{0}$$$ in less than three operations, so the answer is $$$3$$$. | 1 |
Codeforces_test | null | 407 | stdin | [
"3\n4 4 2\n1 2 2\n2 4 2\n1 3 4\n3 4 1\n1 4 2\n2 3 1\n6 7 3\n1 2 10\n2 3 3\n3 4 9\n4 5 2\n5 6 1\n2 4 10\n4 6 10\n1 6 3\n1 6 2\n2 4 1\n11 17 10\n1 4 5\n1 3 19\n1 2 10\n3 2 13\n4 5 1\n4 6 11\n3 5 9\n3 6 18\n2 7 17\n5 8 15\n5 10 8\n6 9 4\n7 10 20\n7 8 16\n8 11 3\n9 11 6\n10 11 14\n3 11 1\n3 11 3\n1 11 1\n1 11 4\n1 11 3\n8 2 2\n10 4 1\n3 9 2\n3 9 1\n6 7 3\n"
] | [
"1 2\n2 9 9\n11 3 11 1 3 10 8 4 11 4\n"
] | [
"3\n4 4 2\n1 2 2\n2 4 2\n1 3 4\n3 4 1\n1 4 2\n2 3 1\n6 7 3\n1 2 10\n2 3 3\n3 4 9\n4 5 2\n5 6 1\n2 4 10\n4 6 10\n1 6 3\n1 6 2\n2 4 1\n11 17 10\n1 4 5\n1 3 19\n1 2 10\n3 2 13\n4 5 1\n4 6 11\n3 5 9\n3 6 18\n2 7 17\n5 8 15\n5 10 8\n6 9 4\n7 10 20\n7 8 16\n8 11 3\n9 11 6\n10 11 14\n3 11 1\n3 11 3\n1 11 1\n1 11 4\n1 11 3\n8 2 2\n10 4 1\n3 9 2\n3 9 1\n6 7 3\n",
"1\n3 2 2\n1 2 1000000000\n1 3 1\n1 2 1\n1 3 1\n"
] | [
"1 2\n2 9 9\n11 3 11 1 3 10 8 4 11 4\n",
"1000000000 1\n"
] | This is the hard version of the problem. The difference between the versions is that in this version, there is no additional constraint on $$$m$$$. You can hack only if you solved all versions of this problem.
Recently, the instructors of "T-generation" needed to create a training contest. They were missing one problem, and there was not a single problem on graphs in the contest, so they came up with the following problem.
You are given a connected weighted undirected graph with $$$n$$$ vertices and $$$m$$$ edges, which does not contain self-loops or multiple edges.
There are $$$q$$$ queries of the form $$$(a, b, k)$$$: among all paths from vertex $$$a$$$ to vertex $$$b$$$, find the smallest $$$k$$$-th maximum weight of edges on the path$$$^{\dagger}$$$.
The instructors thought that the problem sounded very interesting, but there is one catch. They do not know how to solve it. Help them and solve the problem, as there are only a few hours left until the contest starts.
$$$^{\dagger}$$$ Let $$$w_1 \ge w_2 \ge \ldots \ge w_{h}$$$ be the weights of all edges in a path, in non-increasing order. The $$$k$$$-th maximum weight of the edges on this path is $$$w_{k}$$$.
Input format:
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. The description of the test cases follows.
The first line of each set of test case contains three integers $$$n, m$$$ and $$$q$$$ ($$$2 \le n \le 400$$$, $$$n - 1 \le m \le \frac{n \cdot (n - 1)}{2}$$$, $$$1 \le q \le 3 \cdot 10^5$$$) — the number of vertices, the number of edges, and the number of questions, respectively.
Each of the following $$$m$$$ lines of each set of test case contains three integers $$$v, u$$$ and $$$w$$$ ($$$1 \le v, u \le n$$$, $$$1 \le w \le 10^9$$$) — the ends of the next edge of the graph and its weight, respectively. It is guaranteed that the graph does not contain self-loops and multiple edges.
Each of the following $$$q$$$ lines of each set of test case contains three integers $$$a, b$$$ and $$$k$$$ ($$$1 \le a, b \le n$$$, $$$k \ge 1$$$) — the next question. It is guaranteed that any path from vertex $$$a$$$ to vertex $$$b$$$ contains at least $$$k$$$ edges.
It is guaranteed that the sum of the values of $$$n$$$ across all sets of test cases does not exceed $$$400$$$.
It is guaranteed that the sum of the values of $$$q$$$ across all sets of test cases does not exceed $$$3 \cdot 10^5$$$.
Output format:
For each set of test case, output the answers to all questions.
Example Input 0:
3
4 4 2
1 2 2
2 4 2
1 3 4
3 4 1
1 4 2
2 3 1
6 7 3
1 2 10
2 3 3
3 4 9
4 5 2
5 6 1
2 4 10
4 6 10
1 6 3
1 6 2
2 4 1
11 17 10
1 4 5
1 3 19
1 2 10
3 2 13
4 5 1
4 6 11
3 5 9
3 6 18
2 7 17
5 8 15
5 10 8
6 9 4
7 10 20
7 8 16
8 11 3
9 11 6
10 11 14
3 11 1
3 11 3
1 11 1
1 11 4
1 11 3
8 2 2
10 4 1
3 9 2
3 9 1
6 7 3
Example Output 0:
1 2
2 9 9
11 3 11 1 3 10 8 4 11 4
Notes:
In the first set of test cases, one of the optimal paths in the first query is the path $$$1 \rightarrow 3 \rightarrow 4$$$; the $$$2$$$-nd maximum weight of the edges on this path is $$$1$$$. In the second query, one of the optimal paths is $$$2 \rightarrow 4 \rightarrow 3$$$; $$$1$$$-st maximum weight of the edges is $$$2$$$.
In the second set of input data, one of the optimal paths in the first query is the path $$$1 \rightarrow 2 \rightarrow 4 \rightarrow 5 \rightarrow 6$$$; the $$$3$$$-rd maximum weight of the edges on this path is $$$2$$$. | 1 |
Codeforces_test | null | 408 | stdin | [
"3\n4\n#...\n.#..\n..#.\n...#\n2\n.#..\n.#..\n1\n...#\n"
] | [
"4 3 2 1 \n2 2 \n4\n"
] | [
"3\n4\n#...\n.#..\n..#.\n...#\n2\n.#..\n.#..\n1\n...#\n"
] | [
"4 3 2 1 \n2 2 \n4 \n"
] | You are playing your favorite rhythm game, osu!mania. The layout of your beatmap consists of $$$n$$$ rows and $$$4$$$ columns. Because notes at the bottom are closer, you will process the bottommost row first and the topmost row last. Each row will contain exactly one note, represented as a '#'.
For each note $$$1, 2, \dots, n$$$, in the order of processing, output the column in which the note appears.
Input format:
The first line contains $$$t$$$ ($$$1 \leq t \leq 100$$$) — the number of test cases.
For each test case, the first line contains $$$n$$$ ($$$1 \leq n \leq 500$$$) — the number of rows of the beatmap.
The following $$$n$$$ lines contain $$$4$$$ characters. The $$$i$$$-th line represents the $$$i$$$-th row of the beatmap from the top. It is guaranteed that the characters are either '.' or '#', and exactly one of the characters is '#'.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$500$$$.
Output format:
For each test case, output $$$n$$$ integers on a new line, the column that the $$$i$$$-th note appears in for all $$$i$$$ from $$$1$$$ to $$$n$$$.
Example Input 0:
3
4
#...
.#..
..#.
...#
2
.#..
.#..
1
...#
Example Output 0:
4 3 2 1
2 2
4
| 1 |
Codeforces_test | null | 409 | stdin | [
"4\n3\nAAA\nAJJ\n6\nJAJAJJ\nJJAJAJ\n6\nAJJJAJ\nAJJAAA\n9\nAJJJJAJAJ\nJAAJJJJJA\n"
] | [
"2\n2\n3\n2\n"
] | [
"4\n3\nAAA\nAJJ\n6\nJAJAJJ\nJJAJAJ\n6\nAJJJAJ\nAJJAAA\n9\nAJJJJAJAJ\nJAAJJJJJA\n"
] | [
"2\n2\n3\n2\n"
] | Álvaro and José are the only candidates running for the presidency of Tepito, a rectangular grid of $$$2$$$ rows and $$$n$$$ columns, where each cell represents a house. It is guaranteed that $$$n$$$ is a multiple of $$$3$$$.
Under the voting system of Tepito, the grid will be split into districts, which consist of any $$$3$$$ houses that are connected$$$^{\text{∗}}$$$. Each house will belong to exactly one district.
Each district will cast a single vote. The district will vote for Álvaro or José respectively if at least $$$2$$$ houses in that district select them. Therefore, a total of $$$\frac{2n}{3}$$$ votes will be cast.
As Álvaro is the current president, he knows exactly which candidate each house will select. If Álvaro divides the houses into districts optimally, determine the maximum number of votes he can get.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains one integer $$$n$$$ ($$$3 \le n \le 10^5$$$; $$$n$$$ is a multiple of $$$3$$$) — the number of columns of Tepito.
The following two lines each contain a string of length $$$n$$$. The $$$i$$$-th line contains the string $$$s_i$$$, consisting of the characters $$$\texttt{A}$$$ and $$$\texttt{J}$$$. If $$$s_{i,j}=\texttt{A}$$$, the house in the $$$i$$$-th row and $$$j$$$-th column will select Álvaro. Otherwise if $$$s_{i,j}=\texttt{J}$$$, the house will select José.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.
Output format:
For each test case, output a single integer — the maximum number of districts Álvaro can win by optimally dividing the houses into districts.
Example Input 0:
4
3
AAA
AJJ
6
JAJAJJ
JJAJAJ
6
AJJJAJ
AJJAAA
9
AJJJJAJAJ
JAAJJJJJA
Example Output 0:
2
2
3
2
Notes:
The image below showcases the optimal arrangement of districts Álvaro can use for each test case in the example. | 1 |
Codeforces_test | null | 410 | stdin | [
"5\n1\n1\n2\n1 2\n3\n2 4 6\n4\n1000000000 999999999 999999998 999999997\n10\n3 1 4 1 5 9 2 6 5 3\n"
] | [
"0\n2\n1\n3\n8\n"
] | [
"5\n1\n1\n2\n1 2\n3\n2 4 6\n4\n1000000000 999999999 999999998 999999997\n10\n3 1 4 1 5 9 2 6 5 3\n",
"1\n6\n1000000000 900000000 900000000 800000000 694967296 998046875\n"
] | [
"0\n2\n1\n3\n8\n",
"2\n"
] | To train young Kevin's arithmetic skills, his mother devised the following problem.
Given $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ and a sum $$$s$$$ initialized to $$$0$$$, Kevin performs the following operation for $$$i = 1, 2, \ldots, n$$$ in order:
- Add $$$a_i$$$ to $$$s$$$. If the resulting $$$s$$$ is even, Kevin earns a point and repeatedly divides $$$s$$$ by $$$2$$$ until it becomes odd.
Note that Kevin can earn at most one point per operation, regardless of how many divisions he does.
Since these divisions are considered more beneficial for Kevin's development, his mother wants to rearrange $$$a$$$ so that the number of Kevin's total points is maximized. Determine the maximum number of points.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 500$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1\leq n \leq 100$$$) — the number of integers.
The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1\leq a_i \leq 10^9$$$).
Output format:
For each test case, output one integer — the maximum number of points.
Example Input 0:
5
1
1
2
1 2
3
2 4 6
4
1000000000 999999999 999999998 999999997
10
3 1 4 1 5 9 2 6 5 3
Example Output 0:
0
2
1
3
8
Notes:
In the first test case, the only arrangement of $$$a$$$ is $$$[1]$$$. $$$s$$$ becomes $$$1$$$. Kevin earns no points.
In the second test case, the only possible arrangement of $$$a$$$ is $$$[2, 1]$$$. $$$s$$$ becomes $$$1$$$ and $$$1$$$ successively. Kevin earns points in both operations.
In the third test case, one possible arrangement of $$$a$$$ is $$$[2, 4, 6]$$$. $$$s$$$ becomes $$$1$$$, $$$5$$$, and $$$11$$$ successively. Kevin earns a point in the first operation. | 1 |
Codeforces_test | null | 411 | stdin | [
"4\n3\n5 4 5\n3\n4 5 4\n10\n3 3 3 3 4 1 2 3 4 5\n9\n17 89 92 42 29 92 14 70 45\n"
] | [
"7\n6\n10\n97\n"
] | [
"4\n3\n5 4 5\n3\n4 5 4\n10\n3 3 3 3 4 1 2 3 4 5\n9\n17 89 92 42 29 92 14 70 45\n"
] | [
"7\n6\n10\n97\n"
] | You are given an array $$$a_1, a_2, \ldots, a_n$$$ of positive integers.
You can color some elements of the array red, but there cannot be two adjacent red elements (i.e., for $$$1 \leq i \leq n-1$$$, at least one of $$$a_i$$$ and $$$a_{i+1}$$$ must not be red).
Your score is the maximum value of a red element plus the number of red elements. Find the maximum score you can get.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 500$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the length of the array.
The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 1000$$$) — the given array.
Output format:
For each test case, output a single integer: the maximum possible score you can get after coloring some elements red according to the statement.
Example Input 0:
4
3
5 4 5
3
4 5 4
10
3 3 3 3 4 1 2 3 4 5
9
17 89 92 42 29 92 14 70 45
Example Output 0:
7
6
10
97
Notes:
In the first test case, you can color the array as follows: $$$[\color{red}{5}, 4, \color{red}{5}]$$$. Your score is $$$\max([5, 5]) + \text{size}([5, 5]) = 5+2 = 7$$$. This is the maximum score you can get.
In the second test case, you can color the array as follows: $$$[\color{red}{4}, 5, \color{red}{4}]$$$. Your score is $$$\max([4, 4]) + \text{size}([4, 4]) = 4+2 = 6$$$. This is the maximum score you can get.
In the third test case, you can color the array as follows: $$$[\color{red}{3}, 3, \color{red}{3}, 3, \color{red}{4}, 1, 2, \color{red}{3}, 4, \color{red}{5}]$$$. Your score is $$$\max([3, 3, 4, 3, 5]) + \text{size}([3, 3, 4, 3, 5]) = 5+5 = 10$$$. This is the maximum score you can get. | 1 |
Codeforces_test | null | 412 | stdin | [
"3\n2 1\n10 20\n10\n6 7\n3 1 2 4 5 6\n1\n2\n4\n8\n16\n32\n64\n10 4\n1 2 4 8 16 32 64 128 256 512\n10\n100\n1000\n10000\n"
] | [
"26\n7 8 10 12 19 35 67\n513 560 1011 10001\n"
] | [
"3\n2 1\n10 20\n10\n6 7\n3 1 2 4 5 6\n1\n2\n4\n8\n16\n32\n64\n10 4\n1 2 4 8 16 32 64 128 256 512\n10\n100\n1000\n10000\n"
] | [
"26\n7 8 10 12 19 35 67\n513 560 1011 10001\n"
] | One day, the teachers of "T-generation" decided to instill discipline in the pupils, so they lined them up and made them calculate in order. There are a total of $$$n$$$ pupils, the height of the $$$i$$$-th pupil in line is $$$a_i$$$.
The line is comfortable, if for each $$$i$$$ from $$$1$$$ to $$$n - 1$$$, the following condition holds: $$$a_i \cdot 2 \ge a_{i + 1}$$$. Initially, the line is comfortable.
The teachers do not like that the maximum height in the line is too small, so they want to feed the pupils pizza. You know that when a pupil eats one pizza, their height increases by $$$1$$$. One pizza can only be eaten by only one pupil, but each pupil can eat an unlimited number of pizzas. It is important that after all the pupils have eaten their pizzas, the line is comfortable.
The teachers have $$$q$$$ options for how many pizzas they will order. For each option $$$k_i$$$, answer the question: what is the maximum height $$$\max(a_1, a_2, \ldots, a_n)$$$ that can be achieved if the pupils eat at most $$$k_i$$$ pizzas.
Input format:
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.
The first line of each set of test case contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 5 \cdot 10^4$$$) — the number of pupils and the number of options for how many pizzas the teachers will order.
The second line of each set of test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^9$$$) — the heights of the pupils.It is guaranteed that initially, the line is comfortable.
Each of the following $$$q$$$ lines of each set of input data contains one integer $$$k_i$$$ ($$$1 \le k_i \le 10^9$$$) — the next limit for how many pizzas the pupils can eat.
It is guaranteed that the sum of the values of $$$n$$$ across all sets of input data does not exceed $$$5 \cdot 10^4$$$.
It is guaranteed that the sum of the values of $$$q$$$ across all sets of input data does not exceed $$$5 \cdot 10^4$$$.
Output format:
For each test case, for each limit for how many pizzas the pupils can eat, output the maximum value $$$\max(a_1, a_2, \ldots, a_n)$$$ that can be achieved while ensuring that the line is comfortable.
Example Input 0:
3
2 1
10 20
10
6 7
3 1 2 4 5 6
1
2
4
8
16
32
64
10 4
1 2 4 8 16 32 64 128 256 512
10
100
1000
10000
Example Output 0:
26
7 8 10 12 19 35 67
513 560 1011 10001
Notes:
In the first query of the first set of input data, you can first give $$$3$$$ pizzas to the first pupil, and then give $$$6$$$ pizzas to the second pupil, making the final array $$$[13, 26]$$$ (the line is comfortable since $$$13 \cdot 2 \ge 26$$$), and the maximum element in it is $$$26$$$. | 1 |
Codeforces_test | null | 413 | stdin | [
"3\n4 4 2\n1 2 2\n2 4 2\n1 3 4\n3 4 1\n1 4 2\n2 3 1\n6 7 3\n1 2 10\n2 3 3\n3 4 9\n4 5 2\n5 6 1\n2 4 10\n4 6 10\n1 6 3\n1 6 2\n2 4 1\n11 17 10\n1 4 5\n1 3 19\n1 2 10\n3 2 13\n4 5 1\n4 6 11\n3 5 9\n3 6 18\n2 7 17\n5 8 15\n5 10 8\n6 9 4\n7 10 20\n7 8 16\n8 11 3\n9 11 6\n10 11 14\n3 11 1\n3 11 3\n1 11 1\n1 11 4\n1 11 3\n8 2 2\n10 4 1\n3 9 2\n3 9 1\n6 7 3\n"
] | [
"1 2\n2 9 9\n11 3 11 1 3 10 8 4 11 4\n"
] | [
"3\n4 4 2\n1 2 2\n2 4 2\n1 3 4\n3 4 1\n1 4 2\n2 3 1\n6 7 3\n1 2 10\n2 3 3\n3 4 9\n4 5 2\n5 6 1\n2 4 10\n4 6 10\n1 6 3\n1 6 2\n2 4 1\n11 17 10\n1 4 5\n1 3 19\n1 2 10\n3 2 13\n4 5 1\n4 6 11\n3 5 9\n3 6 18\n2 7 17\n5 8 15\n5 10 8\n6 9 4\n7 10 20\n7 8 16\n8 11 3\n9 11 6\n10 11 14\n3 11 1\n3 11 3\n1 11 1\n1 11 4\n1 11 3\n8 2 2\n10 4 1\n3 9 2\n3 9 1\n6 7 3\n",
"1\n3 2 2\n1 2 1000000000\n1 3 1\n1 2 1\n1 3 1\n"
] | [
"1 2\n2 9 9\n11 3 11 1 3 10 8 4 11 4\n",
"1000000000 1\n"
] | This is the easy version of the problem. The difference between the versions is that in this version, there is an additional constraint on $$$m$$$. You can hack only if you solved all versions of this problem.
Recently, the instructors of "T-generation" needed to create a training contest. They were missing one problem, and there was not a single problem on graphs in the contest, so they came up with the following problem.
You are given a connected weighted undirected graph with $$$n$$$ vertices and $$$m$$$ edges, which does not contain self-loops or multiple edges.
There are $$$q$$$ queries of the form $$$(a, b, k)$$$: among all paths from vertex $$$a$$$ to vertex $$$b$$$, find the smallest $$$k$$$-th maximum weight of edges on the path$$$^{\dagger}$$$.
The instructors thought that the problem sounded very interesting, but there is one catch. They do not know how to solve it. Help them and solve the problem, as there are only a few hours left until the contest starts.
$$$^{\dagger}$$$ Let $$$w_1 \ge w_2 \ge \ldots \ge w_{h}$$$ be the weights of all edges in a path, in non-increasing order. The $$$k$$$-th maximum weight of the edges on this path is $$$w_{k}$$$.
Input format:
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. The description of the test cases follows.
The first line of each set of test case contains three integers $$$n, m$$$ and $$$q$$$ ($$$2 \le n \le 400$$$, $$$n - 1 \le m \le \operatorname{min}(\mathbf{400}, \frac{n \cdot (n - 1)}{2})$$$, $$$1 \le q \le 3 \cdot 10^5$$$) — the number of vertices, the number of edges, and the number of questions, respectively.
Each of the following $$$m$$$ lines of each set of test case contains three integers $$$v, u$$$ and $$$w$$$ ($$$1 \le v, u \le n$$$, $$$1 \le w \le 10^9$$$) — the ends of the next edge of the graph and its weight, respectively. It is guaranteed that the graph does not contain self-loops and multiple edges.
Each of the following $$$q$$$ lines of each set of test case contains three integers $$$a, b$$$ and $$$k$$$ ($$$1 \le a, b \le n$$$, $$$k \ge 1$$$) — the next question. It is guaranteed that any path from vertex $$$a$$$ to vertex $$$b$$$ contains at least $$$k$$$ edges.
It is guaranteed that the sum of the values of $$$n$$$ across all sets of test cases does not exceed $$$400$$$.
It is guaranteed that the sum of the values of $$$m$$$ across all sets of test cases does not exceed $$$400$$$.
It is guaranteed that the sum of the values of $$$q$$$ across all sets of test cases does not exceed $$$3 \cdot 10^5$$$.
Output format:
For each test case, output the answers to all questions.
Example Input 0:
3
4 4 2
1 2 2
2 4 2
1 3 4
3 4 1
1 4 2
2 3 1
6 7 3
1 2 10
2 3 3
3 4 9
4 5 2
5 6 1
2 4 10
4 6 10
1 6 3
1 6 2
2 4 1
11 17 10
1 4 5
1 3 19
1 2 10
3 2 13
4 5 1
4 6 11
3 5 9
3 6 18
2 7 17
5 8 15
5 10 8
6 9 4
7 10 20
7 8 16
8 11 3
9 11 6
10 11 14
3 11 1
3 11 3
1 11 1
1 11 4
1 11 3
8 2 2
10 4 1
3 9 2
3 9 1
6 7 3
Example Output 0:
1 2
2 9 9
11 3 11 1 3 10 8 4 11 4
Notes:
In the first set of test cases, one of the optimal paths in the first query is the path $$$1 \rightarrow 3 \rightarrow 4$$$; the $$$2$$$-nd maximum weight of the edges on this path is $$$1$$$. In the second query, one of the optimal paths is $$$2 \rightarrow 4 \rightarrow 3$$$; $$$1$$$-st maximum weight of the edges is $$$2$$$.
In the second set of input data, one of the optimal paths in the first query is the path $$$1 \rightarrow 2 \rightarrow 4 \rightarrow 5 \rightarrow 6$$$; the $$$3$$$-rd maximum weight of the edges on this path is $$$2$$$. | 1 |
Codeforces_test | null | 414 | stdin | [
"3\n2\n1 2\n4\n1 2\n2 3\n2 4\n7\n1 2\n1 3\n2 4\n4 5\n5 6\n5 7\n"
] | [
"0\n2\n4\n"
] | [
"3\n2\n1 2\n4\n1 2\n2 3\n2 4\n7\n1 2\n1 3\n2 4\n4 5\n5 6\n5 7\n",
"1\n22\n1 2\n3 4\n5 6\n7 8\n9 10\n11 12\n13 14\n15 16\n17 2\n17 4\n17 6\n17 8\n18 10\n18 12\n18 14\n18 16\n17 19\n18 19\n19 20\n19 21\n19 22\n",
"1\n16\n1 2\n1 3\n1 4\n2 5\n2 6\n5 7\n6 8\n3 9\n3 10\n9 11\n10 12\n4 13\n4 14\n13 15\n14 16\n"
] | [
"0\n2\n4\n",
"9\n",
"5\n"
] | Recently, Little John got a tree from his aunt to decorate his house. But as it seems, just one tree is not enough to decorate the entire house. Little John has an idea. Maybe he can remove a few vertices from the tree. That will turn it into more trees! Right?
You are given a tree$$$^{\text{∗}}$$$ of $$$n$$$ vertices. You must perform the following operation exactly twice.
- Select a vertex $$$v$$$;
- Remove all edges incident to $$$v$$$, and also the vertex $$$v$$$.
Please find the maximum number of connected components after performing the operation exactly twice.
Two vertices $$$x$$$ and $$$y$$$ are in the same connected component if and only if there exists a path from $$$x$$$ to $$$y$$$. For clarity, note that the graph with $$$0$$$ vertices has $$$0$$$ connected components by definition.$$$^{\text{†}}$$$
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 2 \cdot 10^5$$$).
Each of the next $$$n-1$$$ lines contains two integers $$$u_i$$$ and $$$v_i$$$, denoting the two vertices connected by an edge ($$$1 \le u_i,v_i \le n$$$, $$$u_i \neq v_i$$$). It is guaranteed that the given edges form a tree.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case, output the maximum number of connected components on a separate line.
Example Input 0:
3
2
1 2
4
1 2
2 3
2 4
7
1 2
1 3
2 4
4 5
5 6
5 7
Example Output 0:
0
2
4
Notes:
On the first test case, removing a vertex twice will make the graph empty. By definition, the number of connected components in the graph with $$$0$$$ vertices is $$$0$$$. Therefore, the answer is $$$0$$$.
On the second test case, removing two vertices $$$1$$$ and $$$2$$$ leaves $$$2$$$ connected components. As it is impossible to make $$$3$$$ connected components with $$$2$$$ vertices, the answer is $$$2$$$.
On the third test case, removing two vertices $$$1$$$ and $$$5$$$ leaves $$$4$$$ connected components, which are $$$\left\{ 2,4\right\}$$$, $$$\left\{ 3\right\}$$$, $$$\left\{ 6\right\}$$$, and $$$\left\{ 7\right\}$$$. It can be shown that it is impossible to make $$$5$$$ connected components. Therefore, the answer is $$$4$$$. | 1 |
Codeforces_test | null | 415 | stdin | [
"6\n0001100111\n0000011111\n010101\n111000\n0101\n0110\n0101\n1010\n011001\n001110\n0\n1\n",
"6\n010101\n?0?0??\n0101\n?0?0\n11100101\n????????\n11100101\n???11?1?\n1000100011\n?11?000?0?\n10101\n?1011\n"
] | [
"1\n3\n1\n-1\n-1\n-1\n",
"2\n-1\n0\n2\n2\n-1\n"
] | [
"6\n0001100111\n0000011111\n010101\n111000\n0101\n0110\n0101\n1010\n011001\n001110\n0\n1\n",
"6\n010101\n?0?0??\n0101\n?0?0\n11100101\n????????\n11100101\n???11?1?\n1000100011\n?11?000?0?\n10101\n?1011\n"
] | [
"1\n3\n1\n-1\n-1\n-1\n",
"2\n-1\n0\n2\n2\n-1\n"
] | This is the hard version of the problem. The difference between the versions is that in this version, string $$$t$$$ consists of '0', '1' and '?'. You can hack only if you solved all versions of this problem.
Kevin has a binary string $$$s$$$ of length $$$n$$$. Kevin can perform the following operation:
- Choose two adjacent blocks of $$$s$$$ and swap them.
A block is a maximal substring$$$^{\text{∗}}$$$ of identical characters. Formally, denote $$$s[l,r]$$$ as the substring $$$s_l s_{l+1} \ldots s_r$$$. A block is $$$s[l,r]$$$ satisfying:
- $$$l=1$$$ or $$$s_l\not=s_{l-1}$$$.
- $$$s_l=s_{l+1} = \ldots = s_{r}$$$.
- $$$r=n$$$ or $$$s_r\not=s_{r+1}$$$.
Adjacent blocks are two blocks $$$s[l_1,r_1]$$$ and $$$s[l_2,r_2]$$$ satisfying $$$r_1+1=l_2$$$.
For example, if $$$s=\mathtt{000}\,\mathbf{11}\,\mathbf{00}\,\mathtt{111}$$$, Kevin can choose the two blocks $$$s[4,5]$$$ and $$$s[6,7]$$$ and swap them, transforming $$$s$$$ into $$$\mathtt{000}\,\mathbf{00}\,\mathbf{11}\,\mathtt{111}$$$.
Given a string $$$t$$$ of length $$$n$$$ consisting of '0', '1' and '?', Kevin wants to determine the minimum number of operations required to perform such that for any index $$$i$$$ ($$$1\le i\le n$$$), if $$$t_i\not=$$$ '?' then $$$s_i=t_i$$$. If it is impossible, output $$$-1$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a string $$$s$$$ consisting of '0' and '1'.
The second line of each test case contains a string $$$t$$$ consisting of '0', '1' and '?'.
It is guaranteed that the lengths of $$$s$$$ and $$$t$$$ are the same.
It is guaranteed that the sum of the length of $$$s$$$ over all test cases will not exceed $$$4\cdot 10^5$$$.
Output format:
For each test case, output one integer — the minimum number of operations required. If it is impossible, output $$$-1$$$.
Example Input 0:
6
0001100111
0000011111
010101
111000
0101
0110
0101
1010
011001
001110
0
1
Example Output 0:
1
3
1
-1
-1
-1
Example Input 1:
6
010101
?0?0??
0101
?0?0
11100101
????????
11100101
???11?1?
1000100011
?11?000?0?
10101
?1011
Example Output 1:
2
-1
0
2
2
-1
Notes:
In the first test case of the first example, the possible way is shown in the statement.
In the second test case of the first example, one possible way could be:
- Swap blocks $$$[2, 2], [3, 3]$$$, $$$s$$$ will become $$$\mathtt{001101}$$$.
- Swap blocks $$$[3, 4], [5, 5]$$$, $$$s$$$ will become $$$\mathtt{000111}$$$.
- Swap blocks $$$[1, 3], [4, 6]$$$, $$$s$$$ will become $$$\mathtt{111000}$$$.
In the first test case of the second example, one possible way could be:
- Swap blocks $$$[1, 1], [2, 2]$$$, $$$s$$$ will become $$$\mathtt{100101}$$$.
- Swap blocks $$$[4, 4], [5, 5]$$$, $$$s$$$ will become $$$\mathtt{100011}$$$. | 1 |
Codeforces_test | null | 416 | stdin | [
"10\n4\n0 1 2 3\n6\n0 0 0 0 0 0\n5\n1 0 1 0 1\n5\n3 1 4 1 5\n4\n3 2 1 0\n7\n9 100 0 89 12 2 3\n4\n0 3 9 0\n7\n0 7 0 2 0 7 0\n1\n0\n2\n0 1\n"
] | [
"1\n0\n2\n1\n1\n2\n1\n2\n0\n1\n"
] | [
"10\n4\n0 1 2 3\n6\n0 0 0 0 0 0\n5\n1 0 1 0 1\n5\n3 1 4 1 5\n4\n3 2 1 0\n7\n9 100 0 89 12 2 3\n4\n0 3 9 0\n7\n0 7 0 2 0 7 0\n1\n0\n2\n0 1\n",
"1\n8\n1 0 1 0 1 0 1 0\n"
] | [
"1\n0\n2\n1\n1\n2\n1\n2\n0\n1\n",
"2\n"
] | Evirir the dragon snuck into a wizard's castle and found a mysterious contraption, and their playful instincts caused them to play with (destroy) it...
Evirir the dragon found an array $$$a_1, a_2, \ldots, a_n$$$ of $$$n$$$ non-negative integers.
In one operation, they can choose a non-empty subarray$$$^{\text{∗}}$$$ $$$b$$$ of $$$a$$$ and replace it with the integer $$$\operatorname{mex}(b)$$$$$$^{\text{†}}$$$. They want to use this operation any number of times to make $$$a$$$ only contain zeros. It can be proven that this is always possible under the problem constraints.
What is the minimum number of operations needed?
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 200$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 50$$$), the length of $$$a$$$.
The second line of each test case contains $$$n$$$ space-separated integers, $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \le a_i \le 100$$$).
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$500$$$.
Output format:
For each test case, output a single integer on a line, the minimum number of operations needed to make $$$a$$$ contain only zeros.
Example Input 0:
10
4
0 1 2 3
6
0 0 0 0 0 0
5
1 0 1 0 1
5
3 1 4 1 5
4
3 2 1 0
7
9 100 0 89 12 2 3
4
0 3 9 0
7
0 7 0 2 0 7 0
1
0
2
0 1
Example Output 0:
1
0
2
1
1
2
1
2
0
1
Notes:
In the first test case, Evirir can choose the subarray $$$b = [1, 2, 3]$$$ and replace it with $$$\operatorname{mex}(1, 2, 3) = 0$$$, changing $$$a$$$ from $$$[0, \underline{1, 2, 3}]$$$ to $$$[0, 0]$$$ (where the chosen subarray is underlined). Therefore, the answer is $$$1$$$.
In the second test case, $$$a$$$ already contains only $$$0$$$s, so no operation is needed.
In the third test case, Evirir can change $$$a$$$ as follows: $$$[1, \underline{0, 1, 0, 1}] \to [\underline{1, 2}] \to [0]$$$. Here, $$$\operatorname{mex}(0, 1, 0, 1) = 2$$$ and $$$\operatorname{mex}(1, 2) = 0$$$.
In the fourth test case, Evirir can choose $$$b$$$ to be the entire array $$$a$$$, changing $$$a$$$ from $$$[\underline{3, 1, 4, 1, 5}]$$$ to $$$[0]$$$. | 1 |
Codeforces_test | null | 417 | stdin | [
"3\n4 2 0\n1 2 3 4\n1 1\n3 6 0\n1 2 3\n1 1 2 3 3 2\n4 6 0\n3 1 4 2\n3 1 1 2 3 4\n"
] | [
"YA\nYA\nTIDAK\n"
] | [
"3\n4 2 0\n1 2 3 4\n1 1\n3 6 0\n1 2 3\n1 1 2 3 3 2\n4 6 0\n3 1 4 2\n3 1 1 2 3 4\n"
] | [
"YA\nYA\nTIDAK\n"
] | This is the easy version of the problem. In the two versions, the constraints on $$$q$$$ and the time limit are different. In this version, $$$q=0$$$. You can make hacks only if all the versions of the problem are solved.
A team consisting of $$$n$$$ members, numbered from $$$1$$$ to $$$n$$$, is set to present a slide show at a large meeting. The slide show contains $$$m$$$ slides.
There is an array $$$a$$$ of length $$$n$$$. Initially, the members are standing in a line in the order of $$$a_1, a_2, \ldots, a_n$$$ from front to back. The slide show will be presented in order from slide $$$1$$$ to slide $$$m$$$. Each section will be presented by the member at the front of the line. After each slide is presented, you can move the member at the front of the line to any position in the lineup (without changing the order of the rest of the members). For example, suppose the line of members is $$$[\color{red}{3},1,2,4]$$$. After member $$$3$$$ presents the current slide, you can change the line of members into either $$$[\color{red}{3},1,2,4]$$$, $$$[1,\color{red}{3},2,4]$$$, $$$[1,2,\color{red}{3},4]$$$ or $$$[1,2,4,\color{red}{3}]$$$.
There is also an array $$$b$$$ of length $$$m$$$. The slide show is considered good if it is possible to make member $$$b_i$$$ present slide $$$i$$$ for all $$$i$$$ from $$$1$$$ to $$$m$$$ under these constraints.
However, your annoying boss wants to make $$$q$$$ updates to the array $$$b$$$. In the $$$i$$$-th update, he will choose a slide $$$s_i$$$ and a member $$$t_i$$$ and set $$$b_{s_i} := t_i$$$. Note that these updates are persistent, that is changes made to the array $$$b$$$ will apply when processing future updates.
For each of the $$$q+1$$$ states of array $$$b$$$, the initial state and after each of the $$$q$$$ updates, determine if the slideshow is good.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains three integers $$$n$$$, $$$m$$$ and $$$q$$$ ($$$1 \le n, m \le 2 \cdot 10^5$$$; $$$q=0$$$) — the number of members, the number of sections and the number of updates.
The second line of each test case contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \le a_i \le n$$$) — the initial order of the members from front to back. It is guaranteed that each integer from $$$1$$$ to $$$n$$$ appears exactly once in $$$a$$$.
The third line of each test case contains $$$m$$$ integers $$$b_1, b_2, \ldots, b_m$$$ ($$$1 \le b_i \le n$$$) — the members who should present each section.
It is guaranteed that the sum of $$$n$$$ and the sum of $$$m$$$ over all test cases do not exceed $$$2 \cdot 10^5$$$ respectively.
Output format:
For each test case, output $$$q+1$$$ lines corresponding to the $$$q+1$$$ states of the array $$$b$$$. Output "YA" if the slide show is good, and "TIDAK" otherwise.
You can output the answer in any case (upper or lower). For example, the strings "yA", "Ya", "ya", and "YA" will be recognized as positive responses.
Example Input 0:
3
4 2 0
1 2 3 4
1 1
3 6 0
1 2 3
1 1 2 3 3 2
4 6 0
3 1 4 2
3 1 1 2 3 4
Example Output 0:
YA
YA
TIDAK
Notes:
For the first test case, you do not need to move the members as both slides are presented by member $$$1$$$, who is already at the front of the line.
For the second test case, the following is a possible way to move members so that the presentation is good:
1. $$$[1,2,3]$$$, do not move member $$$1$$$.
2. $$$[1,2,3]$$$, move member $$$1$$$ after member $$$3$$$.
3. $$$[2,3,1]$$$, move member $$$2$$$ after member $$$3$$$.
4. $$$[3,2,1]$$$, do not move member $$$3$$$.
5. $$$[3,2,1]$$$, move member $$$3$$$ after member $$$1$$$.
6. $$$[2,1,3]$$$, do not move member $$$2$$$. | 1 |
Codeforces_test | null | 418 | stdin | [
"5\n2 1\n1 1\n2 2\n1 2\n3 4\n2 1 3\n10 50\n1 1 3 8 8 9 12 13 27 27\n2 1000000000\n1000000000 500000000\n"
] | [
"1\n2\n5\n53\n1000000000\n"
] | [
"5\n2 1\n1 1\n2 2\n1 2\n3 4\n2 1 3\n10 50\n1 1 3 8 8 9 12 13 27 27\n2 1000000000\n1000000000 500000000\n"
] | [
"1\n2\n5\n53\n1000000000\n"
] | There is a vending machine that sells lemonade. The machine has a total of $$$n$$$ slots. You know that initially, the $$$i$$$-th slot contains $$$a_i$$$ cans of lemonade. There are also $$$n$$$ buttons on the machine, each button corresponds to a slot, with exactly one button corresponding to each slot. Unfortunately, the labels on the buttons have worn off, so you do not know which button corresponds to which slot.
When you press the button corresponding to the $$$i$$$-th slot, one of two events occurs:
- If there is a can of lemonade in the $$$i$$$-th slot, it will drop out and you will take it. At this point, the number of cans in the $$$i$$$-th slot decreases by $$$1$$$.
- If there are no cans of lemonade left in the $$$i$$$-th slot, nothing will drop out.
After pressing, the can drops out so quickly that it is impossible to track from which slot it fell. The contents of the slots are hidden from your view, so you cannot see how many cans are left in each slot. The only thing you know is the initial number of cans in the slots: $$$a_1, a_2, \ldots, a_n$$$.
Determine the minimum number of button presses needed to guarantee that you receive at least $$$k$$$ cans of lemonade.
Note that you can adapt your strategy during the button presses based on whether you received a can or not. It is guaranteed that there are at least $$$k$$$ cans of lemonade in total in the machine. In other words, $$$k \leq a_1 + a_2 + \ldots + a_n$$$.
Input format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 2 \cdot 10^5$$$, $$$1 \leq k \leq 10^9$$$) — the number of slots in the machine and the required number of cans of lemonade.
The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^9$$$) — the number of cans in the slots.
It is guaranteed that $$$k \leq a_1 + a_2 + \ldots + a_n$$$, meaning there are at least $$$k$$$ cans of lemonade in the machine.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case, output a single integer — the minimum number of button presses needed to guarantee that you receive at least $$$k$$$ cans of lemonade.
Example Input 0:
5
2 1
1 1
2 2
1 2
3 4
2 1 3
10 50
1 1 3 8 8 9 12 13 27 27
2 1000000000
1000000000 500000000
Example Output 0:
1
2
5
53
1000000000
Notes:
In the first test case, we can simply press the first button and receive one can of lemonade.
In the second test case, we can press each button once and guarantee that we receive $$$2$$$ cans of lemonade. Note that if we simply press one button twice, we might not be lucky, and that button could correspond to the first slot, in which case we would only receive $$$1$$$ can of lemonade for two presses.
In the third test case, one of the optimal strategies is as follows:
Press the first button twice. After the first press, a can of lemonade will definitely drop out. Then there are two options:
- If no can of lemonade drops after the second press, we know that this button must correspond to the second slot, since $$$a_2 = 1$$$ and $$$a_1, a_3 > 1$$$. Then we can press the second button twice and the third button once. Since $$$a_1, a_3 \geq 2$$$, we will definitely receive three cans of lemonade for these three presses. Thus, after $$$5$$$ presses, we will have $$$4$$$ cans of lemonade.
- If a can of lemonade drops after the second press, we can make one press on the second button and one press on the third button. After each of these presses, we will definitely receive a can of lemonade. Thus, after $$$4$$$ presses, we will have $$$4$$$ cans of lemonade.
It can be shown that it is impossible to guarantee receiving $$$4$$$ cans of lemonade with only $$$4$$$ presses, so the answer is $$$5$$$. | 1 |
Codeforces_test | null | 419 | stdin | [
"3\n5\n1 2 3 4 5\n8\n8 8 5 3 4 6 8 12\n4\n3 3 3 4\n"
] | [
"4\nAmbiguous\n3\n"
] | [
"3\n5\n1 2 3 4 5\n8\n8 8 5 3 4 6 8 12\n4\n3 3 3 4\n"
] | [
"4\nAmbiguous\n3\n"
] | Jane has decided to solve a list of $$$n$$$ problems on Codeforces. The $$$i$$$-th problem in her list has difficulty $$$d_i$$$, and the last problem in the list is the hardest one (for every problem $$$j < n$$$, $$$d_j < d_n$$$).
Jane's problem-solving skill is some integer $$$x$$$ (unknown to you). If a problem's difficulty is greater than $$$x$$$, then Jane cannot solve it, otherwise she can solve it.
Jane has solved all problems form the list, except for the last one — she found out that it was too difficult for her. Can you uniquely determine the value of $$$x$$$ — Jane's problem solving skill?
Input format:
The first line contains one integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases.
Each test case consists of two lines:
- the first line contains one integer $$$n$$$ ($$$2 \le n \le 50$$$) — the number of problems;
- the second line contains $$$n$$$ integers $$$d_1, d_2, \dots, d_n$$$ ($$$1 \le d_i \le 50$$$).
Additional constraint on the input: in every test case, the last problem is more difficult than every other problem (i. e. $$$d_n > d_j$$$ for every $$$j < n$$$). This means that at least one possible value of $$$x$$$ exists.
Output format:
For each test case, print one line:
- if you can determine the value of $$$x$$$ uniquely, print $$$x$$$;
- otherwise, print Ambiguous. The checking program is case-sensitive, so if you print ambiguous or AMBIGUOUS, your answer will be considered wrong.
Example Input 0:
3
5
1 2 3 4 5
8
8 8 5 3 4 6 8 12
4
3 3 3 4
Example Output 0:
4
Ambiguous
3
Notes:
In the second test case of the example, the value of $$$x$$$ could be $$$11$$$, but it also could be $$$10$$$ (other possible values for $$$x$$$ also exist). | 1 |
Codeforces_test | null | 420 | stdin | [
"2\n5\n10\n"
] | [
"1 5 2 4 3\n1 2 10 9 7 4 8 3 6 5\n"
] | [
"2\n5\n10\n",
"99\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60\n61\n62\n63\n64\n65\n66\n67\n68\n69\n70\n71\n72\n73\n74\n75\n76\n77\n78\n79\n80\n81\n82\n83\n84\n85\n86\n87\n88\n89\n90\n91\n92\n93\n94\n95\n96\n97\n98\n99\n100\n",
"99\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n100\n"
] | [
"1 5 4 3 2\n1 10 9 8 7 6 5 4 3 2\n",
"1 2 \n1 3 2 \n1 4 3 2 \n1 5 4 3 2 \n1 6 5 4 3 2 \n1 7 6 5 4 3 2 \n1 8 7 6 5 4 3 2 \n1 9 8 7 6 5 4 3 2 \n1 10 9 8 7 6 5 4 3 2 \n1 11 10 9 8 7 6 5 4 3 2 \n1 12 11 10 9 8 7 6 5 4 3 2 \n1 13 12 11 10 9 8 7 6 5 4 3 2 \n1 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n",
"1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n1 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 \n"
] | Yes, this is another one of those constructive permutation problems.
You are given an integer $$$n$$$. You have to construct a permutation $$$p$$$ of size $$$n$$$, i. e. an array of $$$n$$$ integers, where every integer from $$$1$$$ to $$$n$$$ appears exactly once.
Every pair of adjacent elements in the permutation ($$$p_i$$$ and $$$p_{i+1}$$$) must meet the following condition:
- if one of them is divisible by the other, the condition $$$p_i < p_{i+1}$$$ must hold;
- otherwise, the condition $$$p_i > p_{i+1}$$$ must hold.
Input format:
The first line contains one integer $$$t$$$ ($$$1 \le t \le 99$$$) — the number of test cases.
Each test case consists of one line, containing one integer $$$n$$$ ($$$2 \le n \le 100$$$).
Output format:
For each test case, print the answer as follows:
- if no permutation of size $$$n$$$ meeting the conditions from the statement exists, print $$$-1$$$;
- otherwise, print $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ — the required permutation. If there are mutliple answers, print any of them.
Example Input 0:
2
5
10
Example Output 0:
1 5 2 4 3
1 2 10 9 7 4 8 3 6 5
| 1 |
Codeforces_test | null | 421 | stdin | [
"8\n2\n114 109\n2\n17 10\n3\n76 83 88\n8\n38 45 38 80 85 92 99 106\n5\n63 58 65 58 65\n8\n117 124 48 53 48 43 54 49\n5\n95 102 107 114 121\n10\n72 77 82 75 70 75 68 75 68 75\n"
] | [
"YES\nYES\nYES\nNO\nYES\nNO\nYES\nYES\n"
] | [
"8\n2\n114 109\n2\n17 10\n3\n76 83 88\n8\n38 45 38 80 85 92 99 106\n5\n63 58 65 58 65\n8\n117 124 48 53 48 43 54 49\n5\n95 102 107 114 121\n10\n72 77 82 75 70 75 68 75 68 75\n",
"19\n38\n1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 121 126 121 126 121 126 121 126 121 126 121 126\n2\n48 1\n2\n48 53\n2\n48 59\n2\n20 48\n2\n43 48\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n2\n1 2\n"
] | [
"YES\nYES\nYES\nNO\nYES\nNO\nYES\nYES\n",
"YES\nNO\nYES\nNO\nNO\nYES\nNO\nNO\nNO\nNO\nNO\nNO\nNO\nNO\nNO\nNO\nNO\nNO\nNO\n"
] | Boris Notkin composes melodies. He represents them as a sequence of notes, where each note is encoded as an integer from $$$0$$$ to $$$127$$$ inclusive. The interval between two notes $$$a$$$ and $$$b$$$ is equal to $$$|a - b|$$$ semitones.
Boris considers a melody perfect if the interval between each two adjacent notes is either $$$5$$$ semitones or $$$7$$$ semitones.
After composing his latest melodies, he enthusiastically shows you his collection of works. Help Boris Notkin understand whether his melodies are perfect.
Input format:
The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of melodies.
Each melody is described by two lines.
The first line contains an integer $$$n$$$ ($$$2 \leq n \leq 50$$$) — the number of notes in the melody.
The second line contains $$$n$$$ integers $$$a_{1}, a_{2}, \dots, a_{n}$$$ ($$$0 \leq a_{i} \leq 127$$$) — the notes of the melody.
Output format:
For each melody, output "YES", if it is perfect; otherwise, output "NO".
You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.
Example Input 0:
8
2
114 109
2
17 10
3
76 83 88
8
38 45 38 80 85 92 99 106
5
63 58 65 58 65
8
117 124 48 53 48 43 54 49
5
95 102 107 114 121
10
72 77 82 75 70 75 68 75 68 75
Example Output 0:
YES
YES
YES
NO
YES
NO
YES
YES
| 1 |
Codeforces_test | null | 422 | stdin | [
"3\n5 1 1\n10101\n5 2 1\n10101\n6 3 2\n000000\n"
] | [
"2\n0\n1\n"
] | [
"3\n5 1 1\n10101\n5 2 1\n10101\n6 3 2\n000000\n"
] | [
"2\n0\n1\n"
] | Rostam's loyal horse, Rakhsh, has seen better days. Once powerful and fast, Rakhsh has grown weaker over time, struggling to even move. Rostam worries that if too many parts of Rakhsh's body lose strength at once, Rakhsh might stop entirely. To keep his companion going, Rostam decides to strengthen Rakhsh, bit by bit, so no part of his body is too frail for too long.
Imagine Rakhsh's body as a line of spots represented by a binary string $$$s$$$ of length $$$n$$$, where each $$$0$$$ means a weak spot and each $$$1$$$ means a strong one. Rostam's goal is to make sure that no interval of $$$m$$$ consecutive spots is entirely weak (all $$$0$$$s).
Luckily, Rostam has a special ability called Timar, inherited from his mother Rudabeh at birth. With Timar, he can select any segment of length $$$k$$$ and instantly strengthen all of it (changing every character in that segment to $$$1$$$). The challenge is to figure out the minimum number of times Rostam needs to use Timar to keep Rakhsh moving, ensuring there are no consecutive entirely weak spots of length $$$m$$$.
Input format:
The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$), the number of test cases.
The first line of each test case contains three numbers $$$n$$$, $$$m$$$, $$$k$$$ ($$$1 \le m, k \le n \le 2 \cdot 10^5$$$). The second line of each test case contains a binary string $$$s$$$ of $$$n$$$ characters $$$s_1s_2 \ldots s_n$$$. ($$$s_i \in \{$$$0,1$$$\}$$$ for $$$1 \le i \le n$$$).
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case, output the minimum number of times Rostam needs to use Timar to keep Rakhsh moving, ensuring there are no consecutive entirely weak spots of length $$$m$$$.
Example Input 0:
3
5 1 1
10101
5 2 1
10101
6 3 2
000000
Example Output 0:
2
0
1
Notes:
In the first test case, we should apply an operation on each 0.
In the second test case, $$$s$$$ is already ok.
In the third test case, we can perform an operation on interval $$$[3,4]$$$ to get 001100. | 1 |
Codeforces_test | null | 423 | stdin | [
"5\n1 1\n-179\n1 1\n5 3\n-2 2 -1 3 -1\n2 4\n1 5\n1 3\n7 1\n1 1 1 -4 1 1 1\n1 7\n7 2\n2 -2 2 -2 1 2 -1\n1 7\n2 7\n4 4\n1000000000 1000000000 999999999 -1000000000\n2 4\n3 4\n2 3\n1 3\n"
] | [
"-1\n2\n5\n-1\n8\n6\n6\n2\n-1\n1\n2\n"
] | [
"5\n1 1\n-179\n1 1\n5 3\n-2 2 -1 3 -1\n2 4\n1 5\n1 3\n7 1\n1 1 1 -4 1 1 1\n1 7\n7 2\n2 -2 2 -2 1 2 -1\n1 7\n2 7\n4 4\n1000000000 1000000000 999999999 -1000000000\n2 4\n3 4\n2 3\n1 3\n"
] | [
"-1\n2\n5\n-1\n8\n6\n6\n2\n-1\n1\n2\n"
] | In a desert city with a hilly landscape, the city hall decided to level the road surface by purchasing a dump truck. The road is divided into $$$n$$$ sections, numbered from $$$1$$$ to $$$n$$$ from left to right. The height of the surface in the $$$i$$$-th section is equal to $$$a_i$$$. If the height of the $$$i$$$-th section is greater than $$$0$$$, then the dump truck must take sand from the $$$i$$$-th section of the road, and if the height of the $$$i$$$-th section is less than $$$0$$$, the dump truck must fill the pit in the $$$i$$$-th section of the road with sand. It is guaranteed that the initial heights are not equal to $$$0$$$.
When the dump truck is in the $$$i$$$-th section of the road, it can either take away $$$x$$$ units of sand, in which case the height of the surface in the $$$i$$$-th section will decrease by $$$x$$$, or it can fill in $$$x$$$ units of sand (provided that it currently has at least $$$x$$$ units of sand in its bed), in which case the height of the surface in the $$$i$$$-th section of the road will increase by $$$x$$$.
The dump truck can start its journey from any section of the road. Moving to an adjacent section on the left or right takes $$$1$$$ minute, and the time for loading and unloading sand can be neglected. The dump truck has an infinite capacity and is initially empty.
You need to find the minimum time required for the dump truck to level the sand so that the height in each section becomes equal to $$$0$$$. Note that after all movements, the dump truck may still have sand left in its bed. You need to solve this problem independently for the segments numbered from $$$l_i$$$ to $$$r_i$$$. Sand outside the segment cannot be used.
Input format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 3 \cdot 10^5$$$) — the number of sections and the number of queries.
The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$-10^9 \le a_i \le 10^9$$$, $$$a_i \neq 0$$$) — the initial height in each section.
The $$$i$$$-th of the following $$$q$$$ lines contains two integers $$$l_i$$$ and $$$r_i$$$ ($$$1 \le l_i \le r_i \le n$$$) — the boundaries of the segment of sections for which the minimum time needs to be determined.
It is guaranteed that the sum of $$$n$$$ over all test cases and the sum of $$$q$$$ over all test cases do not exceed $$$3 \cdot 10^5$$$.
Output format:
For each query, output the minimum time required to level the sand in the segment $$$[l_i, r_i]$$$, or $$$-1$$$ if it is impossible.
Example Input 0:
5
1 1
-179
1 1
5 3
-2 2 -1 3 -1
2 4
1 5
1 3
7 1
1 1 1 -4 1 1 1
1 7
7 2
2 -2 2 -2 1 2 -1
1 7
2 7
4 4
1000000000 1000000000 999999999 -1000000000
2 4
3 4
2 3
1 3
Example Output 0:
-1
2
5
-1
8
6
6
2
-1
1
2
Notes:
In the first test case, $$$179$$$ units of sand need to be added to the only section. However, there is nowhere to take it from, so this is impossible.
In the second test case:
- In the first query, the dump truck can start its journey at the second section. It can take $$$2$$$ units of sand, after which the height in the second section will become $$$0$$$. Then the dump truck can move to the third section. It can pour $$$1$$$ unit of sand there, after which the height in the third section will become $$$0$$$. Then the dump truck can move to the fourth section. There it can take $$$3$$$ units of sand, after which the height in the fourth section will become $$$0$$$. In total, the dump truck will spend $$$2$$$ minutes on movements.
- In the second query, the dump truck can start its journey at the fourth section. It can take $$$3$$$ units of sand, after which the height in the fourth section will become $$$0$$$. Then the dump truck can move to the fifth section. It can pour $$$1$$$ unit of sand there, after which the height in the fifth section will become $$$0$$$. Then the dump truck can move back to the fourth section and then to the third. It can pour $$$1$$$ unit of sand there, after which the height in the third section will become $$$0$$$. Then the dump truck can move to the second section. It can take $$$2$$$ units of sand. Then it can move to the first section. It can pour $$$2$$$ units of sand there, after which the height in the first section will become $$$0$$$. In total, the dump truck will spend $$$5$$$ minutes on movements.
- In the third query, the dump truck will not be able to make the height in each section equal to $$$0$$$. | 1 |
Codeforces_test | null | 424 | stdin | [
"7\n5 5\n100 20 80\n1 5 30 100\n1 2 20 50\n2 3 20 50\n3 4 20 50\n4 5 20 50\n2 1\n100 50 60\n1 2 55 110\n4 4\n100 40 60\n1 2 30 100\n2 4 30 100\n1 3 20 50\n3 4 20 50\n3 3\n100 80 90\n1 2 1 10\n2 3 10 50\n1 3 20 21\n3 2\n58 55 57\n2 1 1 3\n2 3 3 4\n2 1\n12 9 10\n2 1 6 10\n5 5\n8 5 6\n2 1 1 8\n2 3 4 8\n4 2 2 4\n5 3 3 4\n4 5 2 6\n"
] | [
"0\n-1\n60\n80\n53\n3\n2\n"
] | [
"7\n5 5\n100 20 80\n1 5 30 100\n1 2 20 50\n2 3 20 50\n3 4 20 50\n4 5 20 50\n2 1\n100 50 60\n1 2 55 110\n4 4\n100 40 60\n1 2 30 100\n2 4 30 100\n1 3 20 50\n3 4 20 50\n3 3\n100 80 90\n1 2 1 10\n2 3 10 50\n1 3 20 21\n3 2\n58 55 57\n2 1 1 3\n2 3 3 4\n2 1\n12 9 10\n2 1 6 10\n5 5\n8 5 6\n2 1 1 8\n2 3 4 8\n4 2 2 4\n5 3 3 4\n4 5 2 6\n"
] | [
"0\n-1\n60\n80\n53\n3\n2\n"
] | You live in a city consisting of $$$n$$$ intersections and $$$m$$$ streets connecting some pairs of intersections. You can travel in either direction on each street. No two streets connect the same pair of intersections, and no street connects an intersection to itself. You can reach any intersection from any other, possibly passing through some other intersections.
Every minute, you can board a bus at intersection $$$u_i$$$ and travel for $$$l_{i1}$$$ minutes to intersection $$$v_i$$$. Conversely, you can travel from intersection $$$v_i$$$ to intersection $$$u_i$$$ in $$$l_{i1}$$$ minutes. You can only board and exit the bus at intersections. You can only board the bus at an intersection if you are currently there.
You can also walk along each street, which takes $$$l_{i2} > l_{i1}$$$ minutes.
You can make stops at intersections.
You live at intersection number $$$1$$$. Today you woke up at time $$$0$$$, and you have an important event scheduled at intersection number $$$n$$$, which you must reach no later than time $$$t_0$$$. You also have a phone call planned that will last from $$$t_1$$$ to $$$t_2$$$ minutes ($$$t_1 < t_2 < t_0$$$).
During the phone call, you cannot ride the bus, but you can walk along any streets, make stops, or stay at home. You can exit the bus at minute $$$t_1$$$ and board the bus again at minute $$$t_2$$$.
Since you want to get enough sleep, you became curious — how late can you leave home to have time to talk on the phone and still not be late for the event?
Input format:
The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The following are the descriptions of the test cases.
The first line of each test case contains two integers $$$n$$$, $$$m$$$ ($$$2 \le n \le 10^5, 1 \le m \le 10^5$$$) — the number of intersections and streets in the city.
The second line of each test case contains three integers $$$t_0$$$, $$$t_1$$$, $$$t_2$$$ ($$$1 \le t_1 < t_2 < t_0 \le 10^9$$$) — the start time of the event, the start time of the phone call, and its end time, respectively.
The next $$$m$$$ lines of each test case contain descriptions of the streets.
The $$$i$$$-th line contains four integers $$$u_i$$$, $$$v_i$$$, $$$l_{i1}$$$, $$$l_{i2}$$$ ($$$1 \le u_i, v_i \le n$$$, $$$u_i \neq v_i$$$, $$$1 \le l_{i1} < l_{i2} \le 10^9$$$) — the numbers of the intersections connected by the $$$i$$$-th street, as well as the travel time along the street by bus and on foot. It is guaranteed that no two streets connect the same pair of intersections and that it is possible to reach any intersection from any other.
It is guaranteed that the sum of the values of $$$n$$$ across all test cases does not exceed $$$10^5$$$. It is also guaranteed that the sum of the values of $$$m$$$ across all test cases does not exceed $$$10^5$$$.
Output format:
For each test case, output a single integer — the latest time you can leave home to have time to talk on the phone and not be late for the event. If you cannot reach the event on time, output -1.
Example Input 0:
7
5 5
100 20 80
1 5 30 100
1 2 20 50
2 3 20 50
3 4 20 50
4 5 20 50
2 1
100 50 60
1 2 55 110
4 4
100 40 60
1 2 30 100
2 4 30 100
1 3 20 50
3 4 20 50
3 3
100 80 90
1 2 1 10
2 3 10 50
1 3 20 21
3 2
58 55 57
2 1 1 3
2 3 3 4
2 1
12 9 10
2 1 6 10
5 5
8 5 6
2 1 1 8
2 3 4 8
4 2 2 4
5 3 3 4
4 5 2 6
Example Output 0:
0
-1
60
80
53
3
2
| 1 |
Codeforces_test | null | 425 | stdin | [
"3\n2 6\n7 1\n8 5\n"
] | [
"1 3 \n1 3 7 9 \n1 3 5 7 9\n"
] | [
"3\n2 6\n7 1\n8 5\n",
"9\n1000000000 1\n1000000000 2\n1000000000 3\n1000000000 4\n1000000000 5\n1000000000 6\n1000000000 7\n1000000000 8\n1000000000 9\n",
"3\n5388586 7\n1625078 5\n548212 3\n"
] | [
"1 3 \n1 3 7 9 \n1 3 5 7 9 \n",
"1 3 7 9 \n1 3 7 9 \n1 3 7 9 \n1 3 7 9 \n1 3 5 7 9 \n1 3 7 9 \n1 3 7 9 \n1 3 7 9 \n1 3 7 9 \n",
"1 3 7 9 \n1 3 5 7 9 \n1 3 7 9 \n"
] | Artem wrote the digit $$$d$$$ on the board exactly $$$n!$$$ times in a row. So, he got the number $$$dddddd \dots ddd$$$ (exactly $$$n!$$$ digits).
Now he is curious about which odd digits from $$$1$$$ to $$$9$$$ divide the number written on the board.
Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. The next $$$t$$$ test cases follow.
Each test case consists of a single line containing two integers $$$n$$$ and $$$d$$$ ($$$2 \le n \le 10^9$$$, $$$1 \le d \le 9$$$).
Output format:
For each test case, output the odd digits in ascending order that divide the number written on the board.
Example Input 0:
3
2 6
7 1
8 5
Example Output 0:
1 3
1 3 7 9
1 3 5 7 9
Notes:
The factorial of a positive integer $$$n$$$ ($$$n!$$$) is the product of all integers from $$$1$$$ to $$$n$$$. For example, the factorial of $$$5$$$ is $$$1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$$$. | 1 |
Codeforces_test | null | 426 | stdin | [
"9\n5\n1 1 1 2 3\n6\n2 1 2 2 1 1\n4\n1 2 1 1\n6\n2 1 1 2 2 4\n4\n2 1 2 3\n6\n1 2 2 1 2 1\n5\n4 5 5 1 5\n7\n1 4 3 5 1 1 3\n7\n3 1 3 2 2 3 3\n"
] | [
"1\n2\n1\n0\n0\n1\n1\n0\n2\n"
] | [
"9\n5\n1 1 1 2 3\n6\n2 1 2 2 1 1\n4\n1 2 1 1\n6\n2 1 1 2 2 4\n4\n2 1 2 3\n6\n1 2 2 1 2 1\n5\n4 5 5 1 5\n7\n1 4 3 5 1 1 3\n7\n3 1 3 2 2 3 3\n"
] | [
"1\n2\n1\n0\n0\n1\n1\n0\n2\n"
] | Even in university, students need to relax. That is why Sakurakos teacher decided to go on a field trip. It is known that all of the students will be walking in one line. The student with index $$$i$$$ has some topic of interest which is described as $$$a_i$$$. As a teacher, you want to minimise the disturbance of the line of students.
The disturbance of the line is defined as the number of neighbouring people with the same topic of interest. In other words, disturbance is the number of indices $$$j$$$ ($$$1 \le j < n$$$) such that $$$a_j = a_{j + 1}$$$.
In order to do this, you can choose index $$$i$$$ ($$$1\le i\le n$$$) and swap students at positions $$$i$$$ and $$$n-i+1$$$. You can perform any number of swaps.
Your task is to determine the minimal amount of disturbance that you can achieve by doing the operation described above any number of times.
Input format:
The first line contains one integer $$$t$$$ ($$$1\le t\le 10^4$$$) — the number of test cases.
Each test case is described by two lines.
- The first line contains one integer $$$n$$$ ($$$2 \le n \le 10^5$$$) — the length of the line of students.
- The second line contains $$$n$$$ integers $$$a_i$$$ ($$$1\le a_i\le n$$$) — the topics of interest of students in line.
It is guaranteed that the sum of $$$n$$$ across all test cases does not exceed $$$2\cdot 10^5$$$.
Output format:
For each test case, output the minimal possible disturbance of the line that you can achieve.
Example Input 0:
9
5
1 1 1 2 3
6
2 1 2 2 1 1
4
1 2 1 1
6
2 1 1 2 2 4
4
2 1 2 3
6
1 2 2 1 2 1
5
4 5 5 1 5
7
1 4 3 5 1 1 3
7
3 1 3 2 2 3 3
Example Output 0:
1
2
1
0
0
1
1
0
2
Notes:
In the first example, it is necessary to apply the operation to $$$i=2$$$, thus the array will become $$$[1, \textbf{2}, 1, \textbf{1}, 3]$$$, with the bold elements indicating those that have swapped places. The disturbance of this array is equal to $$$1$$$.
In the fourth example, it is sufficient to apply the operation to $$$i=3$$$, thus the array will become $$$[2, 1, \textbf{2}, \textbf{1}, 2, 4]$$$. The disturbance of this array is equal to $$$0$$$.
In the eighth example, it is sufficient to apply the operation to $$$i=3$$$, thus the array will become $$$[1, 4, \textbf{1}, 5, \textbf{3}, 1, 3]$$$. The disturbance of this array is equal to $$$0$$$. | 1 |
Codeforces_test | null | 427 | stdin | [
"4\n1 1\n3 2\n3 3\n15 8\n"
] | [
"1\n1\n3\n1 2 3\n-1\n5\n1 4 7 10 13\n"
] | [
"4\n1 1\n3 2\n3 3\n15 8\n",
"2\n199999 199999\n1 1\n",
"2\n1 1\n199999 1\n",
"1\n199999 62226\n",
"2\n3 1\n199997 108263\n",
"10\n9999 1\n9999 9999\n15673 1\n38721 38721\n89211 1\n30183 30183\n5023 1\n1023 1023\n111 1\n57 57\n",
"10\n57 1\n111 111\n1023 1\n5023 5023\n30183 1\n89211 89211\n38721 1\n15673 15673\n9999 1\n9999 9999\n",
"1\n199999 199998\n"
] | [
"1\n1\n3\n1 2 3\n-1\n3\n1 8 9\n",
"-1\n1\n1\n",
"1\n1\n-1\n",
"3\n1 62226 62227\n",
"-1\n3\n1 108262 108265\n",
"-1\n-1\n-1\n-1\n-1\n-1\n-1\n-1\n-1\n-1\n",
"-1\n-1\n-1\n-1\n-1\n-1\n-1\n-1\n-1\n-1\n",
"3\n1 199998 199999\n"
] | You are given an array $$$a = [1, 2, \ldots, n]$$$, where $$$n$$$ is odd, and an integer $$$k$$$.
Your task is to choose an odd positive integer $$$m$$$ and to split $$$a$$$ into $$$m$$$ subarrays$$$^{\dagger}$$$ $$$b_1, b_2, \ldots, b_m$$$ such that:
- Each element of the array $$$a$$$ belongs to exactly one subarray.
- For all $$$1 \le i \le m$$$, $$$|b_i|$$$ is odd, i.e., the length of each subarray is odd.
- $$$\operatorname{median}([\operatorname{median}(b_1), \operatorname{median}(b_2), \ldots, \operatorname{median}(b_m)]) = k$$$, i.e., the median$$$^{\ddagger}$$$ of the array of medians of all subarrays must equal $$$k$$$. $$$\operatorname{median}(c)$$$ denotes the median of the array $$$c$$$.
$$$^{\dagger}$$$A subarray of the array $$$a$$$ of length $$$n$$$ is the array $$$[a_l, a_{l + 1}, \ldots, a_r]$$$ for some integers $$$1 \le l \le r \le n$$$.
$$$^{\ddagger}$$$A median of the array of odd length is the middle element after the array is sorted in non-decreasing order. For example: $$$\operatorname{median}([1,2,5,4,3]) = 3$$$, $$$\operatorname{median}([3,2,1]) = 2$$$, $$$\operatorname{median}([2,1,2,1,2,2,2]) = 2$$$.
Input format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 5000$$$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k \le n < 2 \cdot 10^5$$$, $$$n$$$ is odd) — the length of array $$$a$$$ and the desired median of the array of medians of all subarrays.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case:
- If there is no suitable partition, output $$$-1$$$ in a single line.
- Otherwise, in the first line, output an odd integer $$$m$$$ ($$$1 \le m \le n$$$), and in the second line, output $$$m$$$ distinct integers $$$p_1, p_2 , p_3 , \ldots, p_m$$$ ($$$1 = p_1 < p_2 < p_3 < \ldots < p_m \le n$$$) — denoting the left borders of each subarray.
In detail, for a valid answer $$$[p_1, p_2, \ldots, p_m]$$$:
- $$$b_1 = \left[ a_{p_1}, a_{p_1 + 1}, \ldots, a_{p_2 - 1} \right]$$$
- $$$b_2 = \left[ a_{p_2}, a_{p_2 + 1}, \ldots, a_{p_3 - 1} \right]$$$
- $$$\ldots$$$
- $$$b_m = \left[ a_{p_m}, a_{p_m + 1}, \ldots, a_n \right]$$$.
If there are multiple solutions, you can output any of them.
Example Input 0:
4
1 1
3 2
3 3
15 8
Example Output 0:
1
1
3
1 2 3
-1
5
1 4 7 10 13
Notes:
In the first test case, the given partition has $$$m = 1$$$ and $$$b_1 = [1]$$$. It is obvious that $$$\operatorname{median}([\operatorname{median}([1])]) = \operatorname{median}([1]) = 1$$$.
In the second test case, the given partition has $$$m = 3$$$ and:
- $$$b_1 = [1]$$$
- $$$b_2 = [2]$$$
- $$$b_3 = [3]$$$
Therefore, $$$\operatorname{median}([\operatorname{median}([1]), \operatorname{median}([2]), \operatorname{median}([3])]) = \operatorname{median}([1, 2, 3]) = 2$$$.
In the third test case, there is no valid partition for $$$k = 3$$$.
In the fourth test case, the given partition has $$$m = 5$$$ and:
- $$$b_1 = [1, 2, 3]$$$
- $$$b_2 = [4, 5, 6]$$$
- $$$b_3 = [7, 8, 9]$$$
- $$$b_4 = [10, 11, 12]$$$
- $$$b_5 = [13, 14, 15]$$$
Therefore, $$$\operatorname{median}([\operatorname{median}([1, 2, 3]), \operatorname{median}([4, 5, 6]), \operatorname{median}([7, 8, 9]), \operatorname{median}([10, 11, 12]), \operatorname{median}([13, 14, 15])]) = \operatorname{median}([2, 5, 8, 11, 14]) = 8$$$. | 1 |
Codeforces_test | null | 428 | stdin | [
"4\n1 2\n3 4\n2 5\n2 5\n3 7\n6 7\n4 5\n2 8\n"
] | [
"1\n3\n2\n3\n"
] | [
"4\n1 2\n3 4\n2 5\n2 5\n3 7\n6 7\n4 5\n2 8\n",
"1\n10 100\n10 100\n"
] | [
"1\n3\n2\n3\n",
"90\n"
] | There are $$$100$$$ rooms arranged in a row and $$$99$$$ doors between them; the $$$i$$$-th door connects rooms $$$i$$$ and $$$i+1$$$. Each door can be either locked or unlocked. Initially, all doors are unlocked.
We say that room $$$x$$$ is reachable from room $$$y$$$ if all doors between them are unlocked.
You know that:
- Alice is in some room from the segment $$$[l, r]$$$;
- Bob is in some room from the segment $$$[L, R]$$$;
- Alice and Bob are in different rooms.
However, you don't know the exact rooms they are in.
You don't want Alice and Bob to be able to reach each other, so you are going to lock some doors to prevent that. What's the smallest number of doors you have to lock so that Alice and Bob cannot meet, regardless of their starting positions inside the given segments?
Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.
The first line of each test case contains two integers $$$l$$$ and $$$r$$$ ($$$1 \le l < r \le 100$$$) — the bounds of the segment of rooms where Alice is located.
The second line of each test case contains two integers $$$L$$$ and $$$R$$$ ($$$1 \le L < R \le 100$$$) — the bounds of the segment of rooms where Bob is located.
Output format:
For each test case, print a single integer — the smallest number of doors you have to lock so that Alice and Bob cannot meet, regardless of their starting positions inside the given segments.
Example Input 0:
4
1 2
3 4
2 5
2 5
3 7
6 7
4 5
2 8
Example Output 0:
1
3
2
3
Notes:
In the first test case, it is sufficient to lock the door between rooms $$$2$$$ and $$$3$$$.
In the second test case, the following doors have to be locked: $$$(2,3)$$$, $$$(3,4)$$$, $$$(4,5)$$$.
In the third test case, the following doors have to be locked: $$$(5, 6)$$$ and $$$(6,7)$$$. | 1 |
Codeforces_test | null | 429 | stdin | [
"4\n5\n2 4 6 2 5\n5\n5 4 4 5 1\n4\n6 8 2 3\n1\n1\n"
] | [
"10\n11\n10\n1\n"
] | [
"4\n5\n2 4 6 2 5\n5\n5 4 4 5 1\n4\n6 8 2 3\n1\n1\n"
] | [
"10\n11\n10\n1\n"
] | You're given an array $$$a$$$ initially containing $$$n$$$ integers. In one operation, you must do the following:
- Choose a position $$$i$$$ such that $$$1 < i \le |a|$$$ and $$$a_i = |a| + 1 - i$$$, where $$$|a|$$$ is the current size of the array.
- Append $$$i - 1$$$ zeros onto the end of $$$a$$$.
After performing this operation as many times as you want, what is the maximum possible length of the array $$$a$$$?
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 1000$$$). The description of the test cases follows.
The first line of each test case contains $$$n$$$ ($$$1 \le n \le 3 \cdot 10^5$$$) — the length of the array $$$a$$$.
The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^{12}$$$).
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.
Output format:
For each test case, output a single integer — the maximum possible length of $$$a$$$ after performing some sequence of operations.
Example Input 0:
4
5
2 4 6 2 5
5
5 4 4 5 1
4
6 8 2 3
1
1
Example Output 0:
10
11
10
1
Notes:
In the first test case, we can first choose $$$i = 4$$$, since $$$a_4 = 5 + 1 - 4 = 2$$$. After this, the array becomes $$$[2, 4, 6, 2, 5, 0, 0, 0]$$$. We can then choose $$$i = 3$$$ since $$$a_3 = 8 + 1 - 3 = 6$$$. After this, the array becomes $$$[2, 4, 6, 2, 5, 0, 0, 0, 0, 0]$$$, which has a length of $$$10$$$. It can be shown that no sequence of operations will make the final array longer.
In the second test case, we can choose $$$i=2$$$, then $$$i=3$$$, then $$$i=4$$$. The final array will be $$$[5, 4, 4, 5, 1, 0, 0, 0, 0, 0, 0]$$$, with a length of $$$11$$$. | 1 |
Codeforces_test | null | 430 | stdin | [
"3\n2 2 4\n1 1 3\n2 1 2\n100 2 1\n4 1 10\n4 4 10\n10 5 2\n1 1 4\n3 1 2\n4 2 5\n2 2 1\n5 3 4\n"
] | [
"Anda\nKamu\nAnda\n"
] | [
"3\n2 2 4\n1 1 3\n2 1 2\n100 2 1\n4 1 10\n4 4 10\n10 5 2\n1 1 4\n3 1 2\n4 2 5\n2 2 1\n5 3 4\n"
] | [
"Anda\nKamu\nAnda\n"
] | Your friends, Anda and Kamu decide to play a game called Grid Game and ask you to become the gamemaster. As the gamemaster, you set up a triangular grid of size $$$N$$$. The grid has $$$N$$$ rows (numbered from $$$1$$$ to $$$N$$$). Row $$$r$$$ has $$$r$$$ cells; the $$$c$$$-th cell of row $$$r$$$ is denoted as $$$(r, c)$$$.
Before the game starts, $$$M$$$ different cells (numbered from $$$1$$$ to $$$M$$$) are chosen: at cell $$$(R_i, C_i)$$$, you add $$$A_i$$$ stones on it. You then give Anda and Kamu an integer $$$K$$$ and commence the game.
Anda and Kamu will take turns alternately with Anda taking the first turn. A player on their turn will do the following.
- Choose a cell $$$(r, c)$$$ with at least one stone on it.
- Remove at least one but at most $$$K$$$ stones from the chosen cell.
- For each cell $$$(x, y)$$$ such that $$$r + 1 \leq x \leq \min(N, r + K)$$$ and $$$c \leq y \leq c + x - r$$$, add zero or more stones but at most $$$K$$$ stones to cell $$$(x, y)$$$.
The following illustrations show all the possible cells in which you can add stones for $$$K = 3$$$. You choose the cell $$$(2, 1)$$$ for the left illustration and the cell $$$(4, 3)$$$ for the right illustration.
A player who is unable to complete their turn (because there are no more stones on the grid) will lose the game, and the opposing player wins. Determine who will win the game if both players play optimally.
Input format:
This problem is a multi-case problem. The first line consists of an integer $$$T$$$ ($$$1 \leq T \leq 100$$$) that represents the number of test cases.
Each test case starts with a single line consisting of three integers $$$N$$$ $$$M$$$ $$$K$$$ ($$$1 \leq N \leq 10^9; 1 \leq M, K \leq 200\,000$$$). Then, each of the next $$$M$$$ lines consists of three integers $$$R_i$$$ $$$C_i$$$ $$$A_i$$$ ($$$1 \leq C_i \leq R_i \leq N; 1 \leq A_1 \leq 10^9$$$). The pairs $$$(R_i, C_i)$$$ are distinct.
The sum of $$$M$$$ across all test cases does not exceed $$$200\,000$$$.
Output format:
For each case, output a string in a single line representing the player who will win the game if both players play optimally. Output Anda if Anda, the first player, wins. Otherwise, output Kamu.
Example Input 0:
3
2 2 4
1 1 3
2 1 2
100 2 1
4 1 10
4 4 10
10 5 2
1 1 4
3 1 2
4 2 5
2 2 1
5 3 4
Example Output 0:
Anda
Kamu
Anda
Notes:
Explanation for the sample input/output #1
For the first case, during the first turn, Anda will remove all the stones from cell $$$(1, 1)$$$ and then add three stones at $$$(2, 1)$$$. The only cell with stones left is now cell $$$(2, 1)$$$ with five stones, so Kamu must remove stones from that cell. No matter how many stones are removed by Kamu, Anda can remove all the remaining stones at $$$(2, 1)$$$ and win the game.
For the second case, Kamu can always mirror whatever move made by Anda until Anda can no longer complete their turn. | 1 |
Codeforces_test | null | 431 | stdin | [
"3 2\n",
"5 3\n"
] | [
"7\n",
"67\n"
] | [
"3 2\n",
"5 3\n",
"1000000 1000000000\n",
"1000000 999999\n",
"1000000 1\n",
"1 1000000000\n",
"644271 497470\n",
"1000000 2\n",
"514514 514\n",
"514514 515515\n",
"514514 514513\n",
"998789 789897\n",
"999888 3\n",
"556888 1324\n",
"51515 55\n",
"8 9\n",
"18 18\n"
] | [
"7\n",
"67\n",
"793302832\n",
"793302830\n",
"1\n",
"1\n",
"189775039\n",
"421273116\n",
"638444431\n",
"915424391\n",
"915424389\n",
"658100689\n",
"117498587\n",
"580017151\n",
"18469295\n",
"2187\n",
"129140163\n"
] | Image generated by ChatGPT 4o.
Alice likes singing. As a singing enthusiast, Alice has listened to countless songs and has tried singing them many times. However, occasionally, some songs make Alice feel bored. After some research, Alice believes that this is because even though the songs she chose are all different, due to her instinctive preference, they all turn out to be musically similar to one another.
To thoroughly analyze this, Alice decided to study the sheet music of the songs. For convenience, Alice represented a song of length $$$n$$$ as an integer sequence $$$a_1, a_2, \ldots, a_n$$$, where $$$a_i$$$ is the pitch of the $$$i$$$-th note. Then she defined the musical equivalence between songs. Two songs $$$a_1, a_2, \ldots, a_n$$$ and $$$b_1, b_2, \ldots, b_n$$$ of length $$$n$$$ are musically equivalent if for all $$$1\leq i<n$$$, both $$$a_i, a_{i+1}$$$ and $$$b_{i}, b_{i+1}$$$ have the same pitch relationship. More specifically, $$$a_i, a_{i+1}$$$ and $$$b_i, b_{i+1}$$$ have the same pitch relationship if either
- $$$a_i < a_{i + 1}$$$ and $$$b_i < b_{i + 1}$$$,
- $$$a_i = a_{i + 1}$$$ and $$$b_i = b_{i + 1}$$$, or
- $$$a_i > a_{i + 1}$$$ and $$$b_i > b_{i + 1}$$$.
Having practiced consistently for a long time, Alice is able to sing any note in the range of $$$[1, k]$$$. She wants to know how many different songs of length $$$n$$$ within her range there are, if we treat musically equivalent songs as the same one. Can you help her calculate the number?
Since the answer might be large, print the answer modulo $$$998244353$$$.
Input format:
The only line contains two integers $$$n, k$$$.
- $$$1\leq n\leq 10^6$$$
- $$$1\leq k \leq 10^9$$$
Output format:
Output the number of different songs modulo $$$998244353$$$.
Example Input 0:
3 2
Example Output 0:
7
Example Input 1:
5 3
Example Output 1:
67
| 1 |
Codeforces_test | null | 432 | stdin | [
"6\n5 2\n3 5\n16 4\n100 3\n6492 10\n10 1\n"
] | [
"2\n3\n1\n4\n21\n10\n"
] | [
"6\n5 2\n3 5\n16 4\n100 3\n6492 10\n10 1\n"
] | [
"2\n3\n1\n4\n21\n10\n"
] | You are given two integers $$$n$$$ and $$$k$$$.
In one operation, you can subtract any power of $$$k$$$ from $$$n$$$. Formally, in one operation, you can replace $$$n$$$ by $$$(n-k^x)$$$ for any non-negative integer $$$x$$$.
Find the minimum number of operations required to make $$$n$$$ equal to $$$0$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The only line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n, k \le 10^9$$$).
Output format:
For each test case, output the minimum number of operations on a new line.
Example Input 0:
6
5 2
3 5
16 4
100 3
6492 10
10 1
Example Output 0:
2
3
1
4
21
10
Notes:
In the first test case, $$$n = 5$$$ and $$$k = 2$$$. We can perform the following sequence of operations:
1. Subtract $$$2^0 = 1$$$ from $$$5$$$. The current value of $$$n$$$ becomes $$$5 - 1 = 4$$$.
2. Subtract $$$2^2 = 4$$$ from $$$4$$$. The current value of $$$n$$$ becomes $$$4 - 4 = 0$$$.
It can be shown that there is no way to make $$$n$$$ equal to $$$0$$$ in less than $$$2$$$ operations. Thus, $$$2$$$ is the answer.
In the second test case, $$$n = 3$$$ and $$$k = 5$$$. We can perform the following sequence of operations:
1. Subtract $$$5^0 = 1$$$ from $$$3$$$. The current value of $$$n$$$ becomes $$$3 - 1 = 2$$$.
2. Subtract $$$5^0 = 1$$$ from $$$2$$$. The current value of $$$n$$$ becomes $$$2 - 1 = 1$$$.
3. Subtract $$$5^0 = 1$$$ from $$$1$$$. The current value of $$$n$$$ becomes $$$1 - 1 = 0$$$.
It can be shown that there is no way to make $$$n$$$ equal to $$$0$$$ in less than $$$3$$$ operations. Thus, $$$3$$$ is the answer. | 1 |
Codeforces_test | null | 433 | stdin | [
"5\n?????\nxbx\nab??e\nabcde\nayy?x\na\nab??e\ndac\npaiu\nmom\n"
] | [
"YES\nxabax\nYES\nabcde\nYES\nayyyx\nNO\nNO\n"
] | [
"5\n?????\nxbx\nab??e\nabcde\nayy?x\na\nab??e\ndac\npaiu\nmom\n"
] | [
"YES\nxbxaa\nYES\nabcde\nYES\nayyax\nNO\nNO\n"
] | Slavic has a very tough exam and needs your help in order to pass it. Here is the question he is struggling with:
There exists a string $$$s$$$, which consists of lowercase English letters and possibly zero or more "?".
Slavic is asked to change each "?" to a lowercase English letter such that string $$$t$$$ becomes a subsequence (not necessarily continuous) of the string $$$s$$$.
Output any such string, or say that it is impossible in case no string that respects the conditions exists.
Input format:
The first line contains a single integer $$$T$$$ ($$$1 \leq T \leq 10^4$$$) — the number of test cases.
The first line of each test case contains a single string $$$s$$$ ($$$1 \leq |s| \leq 2 \cdot 10^5$$$, and $$$s$$$ consists only of lowercase English letters and "?"-s) – the original string you have.
The second line of each test case contains a single string $$$t$$$ ($$$1 \leq |t| \leq |s|$$$, and $$$t$$$ consists only of lowercase English letters) – the string that should be a subsequence of string $$$s$$$.
The sum of $$$|s|$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$, where $$$|x|$$$ denotes the length of the string $$$x$$$.
Output format:
For each test case, if no such string exists as described in the statement, output "NO" (without quotes).
Otherwise, output "YES" (without quotes). Then, output one line — the string that respects all conditions.
You can output "YES" and "NO" in any case (for example, strings "yEs", "yes", and "Yes" will be recognized as a positive response).
If multiple answers are possible, you can output any of them.
Example Input 0:
5
?????
xbx
ab??e
abcde
ayy?x
a
ab??e
dac
paiu
mom
Example Output 0:
YES
xabax
YES
abcde
YES
ayyyx
NO
NO
| 1 |
Codeforces_test | null | 434 | stdin | [
"4\n2 2\n7 2\n5 3\n1000000000 1000000000\n"
] | [
"1\n5\n1\n347369930\n"
] | [
"4\n2 2\n7 2\n5 3\n1000000000 1000000000\n"
] | [
"1\n5\n1\n347369930\n"
] | Klee has an array $$$a$$$ of length $$$n$$$ containing integers $$$[k, k+1, ..., k+n-1]$$$ in that order. Klee wants to choose an index $$$i$$$ ($$$1 \leq i \leq n$$$) such that $$$x = |a_1 + a_2 + \dots + a_i - a_{i+1} - \dots - a_n|$$$ is minimized. Note that for an arbitrary integer $$$z$$$, $$$|z|$$$ represents the absolute value of $$$z$$$.
Output the minimum possible value of $$$x$$$.
Input format:
The first line contains $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.
Each test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \leq n, k \leq 10^9$$$) — the length of the array and the starting element of the array.
Output format:
For each test case, output the minimum value of $$$x$$$ on a new line.
Example Input 0:
4
2 2
7 2
5 3
1000000000 1000000000
Example Output 0:
1
5
1
347369930
Notes:
In the first sample, $$$a = [2, 3]$$$. When $$$i = 1$$$ is chosen, $$$x = |2-3| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.
In the third sample, $$$a = [3, 4, 5, 6, 7]$$$. When $$$i = 3$$$ is chosen, $$$x = |3 + 4 + 5 - 6 - 7| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$. | 1 |
Codeforces_test | null | 435 | stdin | [
"3\n5\n1 2 5 905 2000000\n15\n- 2\n? 2\n? 1\n- 1\n? 1\n+ 4\n+ 2\n? 2\n+ 6\n- 4\n+ 7\n? 2\n? 3\n? 4\n? 2000000\n5\n3 4 5 6 8\n9\n? 5\n- 5\n? 5\n+ 1\n? 2\n- 6\n- 8\n+ 6\n? 5\n5\n6 7 8 9 10\n10\n? 5\n- 6\n? 4\n- 10\n+ 5\n- 8\n+ 3\n+ 2\n- 3\n+ 10\n"
] | [
"2 2 1 6 3 8 8 2000001 \n9 9 9 7 \n1 1\n"
] | [
"3\n5\n1 2 5 905 2000000\n15\n- 2\n? 2\n? 1\n- 1\n? 1\n+ 4\n+ 2\n? 2\n+ 6\n- 4\n+ 7\n? 2\n? 3\n? 4\n? 2000000\n5\n3 4 5 6 8\n9\n? 5\n- 5\n? 5\n+ 1\n? 2\n- 6\n- 8\n+ 6\n? 5\n5\n6 7 8 9 10\n10\n? 5\n- 6\n? 4\n- 10\n+ 5\n- 8\n+ 3\n+ 2\n- 3\n+ 10\n"
] | [
"2 2 1 6 3 8 8 2000001 \n9 9 9 7 \n1 1 \n"
] | Ksyusha decided to start a game development company. To stand out among competitors and achieve success, she decided to write her own game engine. The engine must support a set initially consisting of $$$n$$$ distinct integers $$$a_1, a_2, \ldots, a_n$$$.
The set will undergo $$$m$$$ operations sequentially. The operations can be of the following types:
- Insert element $$$x$$$ into the set;
- Remove element $$$x$$$ from the set;
- Report the $$$k$$$-load of the set.
The $$$k$$$-load of the set is defined as the minimum positive integer $$$d$$$ such that the integers $$$d, d + 1, \ldots, d + (k - 1)$$$ do not appear in this set. For example, the $$$3$$$-load of the set $$$\{3, 4, 6, 11\}$$$ is $$$7$$$, since the integers $$$7, 8, 9$$$ are absent from the set, and no smaller value fits.
Ksyusha is busy with management tasks, so you will have to write the engine. Implement efficient support for the described operations.
Input format:
The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.
The following lines describe the test cases.
The first line contains an integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the initial size of the set.
The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_1 < a_2 < \ldots < a_n \le 2 \cdot 10^6$$$) — the initial state of the set.
The third line contains an integer $$$m$$$ ($$$1 \le m \le 2 \cdot 10^5$$$) — the number of operations.
The next $$$m$$$ lines contain the operations. The operations are given in the following format:
- + $$$x$$$ ($$$1 \le x \le 2 \cdot 10^6$$$) — insert element $$$x$$$ into the set (it is guaranteed that $$$x$$$ is not in the set);
- - $$$x$$$ ($$$1 \le x \le 2 \cdot 10^6$$$) — remove element $$$x$$$ from the set (it is guaranteed that $$$x$$$ is in the set);
- ? $$$k$$$ ($$$1 \le k \le 2 \cdot 10^6$$$) — output the value of the $$$k$$$-load of the set.
It is guaranteed that the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$, and the same holds for $$$m$$$.
Output format:
For each test case, output the answers to the operations of type "?".
Example Input 0:
3
5
1 2 5 905 2000000
15
- 2
? 2
? 1
- 1
? 1
+ 4
+ 2
? 2
+ 6
- 4
+ 7
? 2
? 3
? 4
? 2000000
5
3 4 5 6 8
9
? 5
- 5
? 5
+ 1
? 2
- 6
- 8
+ 6
? 5
5
6 7 8 9 10
10
? 5
- 6
? 4
- 10
+ 5
- 8
+ 3
+ 2
- 3
+ 10
Example Output 0:
2 2 1 6 3 8 8 2000001
9 9 9 7
1 1
| 1 |
Codeforces_test | null | 436 | stdin | [
"3\n4\n\n1\n\n5\n\n1\n\n0\n\n9\n"
] | [
"? 2 3\n\n! 0 0 1\n\n? 2 3\n\n? 2 4\n\n! 0 0 1 2\n\n! 0 0 0 1 3 5 6 7\n"
] | [
"3\n4\n0 0 1\n5\n0 0 1 2\n9\n0 0 0 1 3 5 6 7\n"
] | [
"0 0 1\n0 0 1 2\n0 0 0 1 3 5 6 7\n"
] | This is an interactive problem.
Upon clearing the Waterside Area, Gretel has found a monster named Genokraken, and she's keeping it contained for her scientific studies.
The monster's nerve system can be structured as a tree$$$^{\dagger}$$$ of $$$n$$$ nodes (really, everything should stop resembling trees all the time$$$\ldots$$$), numbered from $$$0$$$ to $$$n-1$$$, with node $$$0$$$ as the root.
Gretel's objective is to learn the exact structure of the monster's nerve system — more specifically, she wants to know the values $$$p_1, p_2, \ldots, p_{n-1}$$$ of the tree, where $$$p_i$$$ ($$$0 \le p_i < i$$$) is the direct parent node of node $$$i$$$ ($$$1 \le i \le n - 1$$$).
She doesn't know exactly how the nodes are placed, but she knows a few convenient facts:
- If we remove root node $$$0$$$ and all adjacent edges, this tree will turn into a forest consisting of only paths$$$^{\ddagger}$$$. Each node that was initially adjacent to the node $$$0$$$ will be the end of some path.
- The nodes are indexed in a way that if $$$1 \le x \le y \le n - 1$$$, then $$$p_x \le p_y$$$.
- Node $$$1$$$ has exactly two adjacent nodes (including the node $$$0$$$).
The tree in this picture does not satisfy the condition, because if we remove node $$$0$$$, then node $$$2$$$ (which was initially adjacent to the node $$$0$$$) will not be the end of the path $$$4-2-5$$$.The tree in this picture does not satisfy the condition, because $$$p_3 \le p_4$$$ must hold.The tree in this picture does not satisfy the condition, because node $$$1$$$ has only one adjacent node.
Gretel can make queries to the containment cell:
- "? a b" ($$$1 \le a, b < n$$$, $$$a \ne b$$$) — the cell will check if the simple path between nodes $$$a$$$ and $$$b$$$ contains the node $$$0$$$.
However, to avoid unexpected consequences by overstimulating the creature, Gretel wants to query at most $$$2n - 6$$$ times. Though Gretel is gifted, she can't do everything all at once, so can you give her a helping hand?
$$$^{\dagger}$$$A tree is a connected graph where every pair of distinct nodes has exactly one simple path connecting them.
$$$^{\ddagger}$$$A path is a tree whose vertices can be listed in the order $$$v_1, v_2, \ldots, v_k$$$ such that the edges are $$$(v_i, v_{i+1})$$$ ($$$1 \le i < k$$$).
Input format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 500$$$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$4 \le n \le 10^4$$$) — the number of nodes in Genokraken's nerve system.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^4$$$.
Example Input 0:
3
4
1
5
1
0
9
Example Output 0:
? 2 3
! 0 0 1
? 2 3
? 2 4
! 0 0 1 2
! 0 0 0 1 3 5 6 7
Notes:
In the first test case, Genokraken's nerve system forms the following tree:
- The answer to "? 2 3" is $$$1$$$. This means that the simple path between nodes $$$2$$$ and $$$3$$$ contains node $$$0$$$.
In the second test case, Genokraken's nerve system forms the following tree:
- The answer to "? 2 3" is $$$1$$$. This means that the simple path between nodes $$$2$$$ and $$$3$$$ contains node $$$0$$$.
- The answer to "? 2 4" is $$$0$$$. This means that the simple path between nodes $$$2$$$ and $$$4$$$ doesn't contain node $$$0$$$.
In the third test case, Genokraken's nerve system forms the following tree: | 1 |
Codeforces_test | null | 437 | stdin | [
"3\n2 1 2\n1 2 1\n",
"4\n5 5 5 5\n0 0 0 0\n",
"5\n1 9 3 7 5\n2 4 6 8 5\n"
] | [
"2 332748119 1\n",
"5 5 5 5\n",
"6 4 3 199648873 2\n"
] | [
"3\n2 1 2\n1 2 1\n",
"4\n5 5 5 5\n0 0 0 0\n",
"5\n1 9 3 7 5\n2 4 6 8 5\n",
"1\n0\n0\n",
"1\n0\n1000000\n",
"1\n1000000\n0\n",
"1\n1000000\n1000000\n",
"20\n443309 129766 7146 25306 61735 943035 371598 498745 993335 611156 910260 422340 664742 667842 954783 553879 829847 999710 772830 428350\n148617 668583 427042 1589 418767 392702 57869 161281 756955 973750 756208 426868 150477 312764 495355 115731 643595 796159 753920 348606\n",
"20\n321380 159267 39407 456869 270423 286924 865039 427905 966965 817491 928742 770903 951860 932419 307057 818599 704973 840018 398306 570917\n926258 470757 193179 794815 12790 412140 177164 152827 820813 640167 757902 582247 11616 666146 774816 646333 195552 120057 332218 20711\n",
"20\n199451 188768 72354 888431 478427 556131 320795 319381 903597 23824 946540 119464 275977 271678 771014 121002 506100 716639 23782 787482\n741581 235248 921633 662722 606814 430893 333457 145058 772989 307268 797281 699943 873441 982530 980278 139250 785193 444640 910516 693501\n",
"20\n40524 180586 67617 394676 724114 862337 851236 210857 840228 230158 965023 393344 525411 536255 123288 422721 344912 556261 612260 892365\n519221 74422 687771 455946 237836 450331 489750 99605 799849 974370 798975 855322 772264 335911 297422 632168 337150 768539 488814 403290\n",
"20\n881597 210087 174561 826238 969802 169227 269993 139332 813859 436493 20503 741907 812529 912515 549561 687441 146039 470566 200737 71930\n333859 839599 453908 286169 831860 469770 609044 54152 826708 678471 838353 973017 596404 652294 576883 162769 926107 55438 104110 1397\n",
"20\n722669 202589 169824 294799 214804 475432 762749 993810 712806 642827 38986 53469 136645 177091 939520 952161 984851 310189 826214 176812\n149183 641773 183046 79393 463567 526207 765337 46383 815883 382571 840047 91397 495228 5676 819345 655687 478064 416336 645410 711186\n",
"20\n600740 232091 202770 726362 423492 818637 256190 922285 649438 849161 20469 439031 423764 516351 328793 216880 859977 224494 414691 319379\n926824 406264 948500 909618 131588 507961 884631 930 842743 49672 841741 246091 394051 322059 136489 149290 30020 740235 223707 383976\n",
"20\n404813 261592 198033 232606 669180 125527 674948 813761 586069 55494 75951 750595 710882 780928 755751 519284 698788 64116 77166 498945\n704463 245438 714637 740526 725612 564398 77922 955478 794919 753773 844120 401471 218191 675441 341951 679207 618977 27134 801321 56081\n",
"20\n282884 291094 230295 664169 877183 431733 205387 668238 559700 261829 94433 61472 998000 120188 145024 747005 499916 941423 665644 603828\n556786 47613 480774 570749 319635 583837 234216 947709 858778 457874 845815 519851 117014 954826 621412 172809 133935 351033 379618 728871\n",
"20\n910419 190865 103374 839463 57835 825789 959256 909104 262613 930809 212704 670190 732007 451822 217994 188049 906417 35625 53724 257830\n972153 81808 175105 393382 850528 952073 119317 12078 826410 719440 569472 937256 78434 858791 378062 250890 598207 840371 794353 824031\n"
] | [
"2 332748119 1 \n",
"5 5 5 5 \n",
"6 4 3 199648873 2 \n",
"0 \n",
"1000000 \n",
"1000000 \n",
"1000000 \n",
"150385282 646686036 580024390 459931971 720047846 40653046 997940504 821058607 405459823 463788016 324476726 740478844 379558797 206919447 271286963 701987283 528912628 709293146 449220549 7146 \n",
"749375091 630961454 438198891 470472600 176732726 185957427 724393763 980321951 857074660 976945820 922112322 622733091 550740500 178641700 473314952 869333050 645415444 462396731 45400 39407 \n",
"949013903 967212384 20526889 680651234 53135948 890291842 452492707 53309450 380369635 896755489 210310968 34186935 910214793 571950882 167421583 71844041 721679632 767212282 199788411 139250 \n",
"399946901 714985804 50265789 152555399 769476855 791530286 926329724 930635100 202505421 245564825 580419256 448910961 574071719 806471808 324984406 849159997 105159587 641055878 449285640 74422 \n",
"50570445 667686507 374214170 56685140 8422140 915847970 215765296 709502943 852141422 371096482 907941005 292222230 547107480 645380342 733725400 303117757 344165471 730323748 399322498 20503 \n",
"350009721 37191804 22186960 897922239 818783644 211873057 820751233 315673802 816244919 619438306 689608643 702696536 983276729 436464897 983743616 538418971 627887178 246975876 40080 38986 \n",
"100479315 872574907 19577361 579214583 397158237 140862651 502629253 733865416 624834124 895730465 177643002 520669180 654320645 366773509 13526975 772076675 616509361 352052264 798625544 20469 \n",
"849181450 173870495 22254735 520102984 26546053 367814002 598260103 252707582 468155469 893042486 865904950 588367894 966658958 532022461 36438921 399150528 722471181 105134942 349441478 55494 \n",
"449883718 893614109 279667711 894873478 116629465 960494679 45993029 47002159 521790045 222551823 344949030 923904394 931108017 340075049 900976948 2758116 311807384 662063997 449275856 61472 \n",
"400032612 672995134 17853059 143440957 320830687 667422109 14042787 670235654 251861650 732315451 262219951 372648782 388024946 71703862 947623492 449000187 489534657 73596507 199687541 35625 \n"
] | Suppose you are working in the Ministry of Digital Development of Berland, and your task is to monitor the industry of video blogging.
There are $$$n$$$ bloggers in Berland. Recently, due to the poor state of the main video platform in Berland, two alternative platforms were introduced. That's why bloggers started to reupload their videos to these alternative platforms. You've got the statistics that the $$$i$$$-th blogger uploaded $$$v_i$$$ videos to the first alternative platform and $$$r_i$$$ videos to the second alternative platform.
You think that a potential user will be upset if even at least one of his favorite bloggers doesn't upload anything. However, if a blogger uploads videos to both platforms, the user will watch that blogger on the platform where more videos are available. So, you've come up with the following function to estimate user experience. Suppose a user watches $$$k$$$ bloggers $$$b_1, b_2, \dots, b_k$$$; then, let user experience be $$$$$$E(b_1, \dots, b_k) = \max\left(\min_{i=1..k}{v[b_i]}, \min_{i=1..k}{r[b_i]}\right).$$$$$$
In order to get some statistics, you want to calculate the value $$$\mathit{avg}_k$$$ that is equal to an average experience among all subsets of bloggers of size $$$k$$$. Also, you have to calculate $$$\mathit{avg}_k$$$ for each $$$k$$$ from $$$1$$$ to $$$n$$$.
Since answers may be too large, print them modulo $$$998\,244\,353$$$.
Input format:
The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of bloggers.
The second line contains $$$n$$$ integers $$$v_1, v_2, \dots, v_n$$$ ($$$0 \le v_i \le 10^6$$$), where $$$v_i$$$ is the number of videos of the $$$i$$$-th blogger on the first alternative platform.
The third line contains $$$n$$$ integers $$$r_1, r_2, \dots, r_n$$$ ($$$0 \le r_i \le 10^6$$$), where $$$r_i$$$ is the number of videos of the $$$i$$$-th blogger on the second alternative platform.
Output format:
Print $$$n$$$ integers $$$\mathit{avg}_1, \mathit{avg}_2, \dots, \mathit{avg}_n$$$.
It can be proven that $$$\mathit{avg}_k$$$ may be represented as an irreducible fraction $$$\dfrac{x}{y}$$$ where $$$y \not\equiv 0 \pmod{998\,244\,353}$$$. So, print $$$\mathit{avg}_k$$$ in a form $$$x \cdot y^{-1} \bmod 998\,244\,353$$$.
Example Input 0:
3
2 1 2
1 2 1
Example Output 0:
2 332748119 1
Example Input 1:
4
5 5 5 5
0 0 0 0
Example Output 1:
5 5 5 5
Example Input 2:
5
1 9 3 7 5
2 4 6 8 5
Example Output 2:
6 4 3 199648873 2
Notes:
In the first example, $$$332748119$$$ is $$$\frac{4}{3}$$$. In the third example, $$$199648873$$$ is $$$\frac{12}{5}$$$. | 1 |
Codeforces_test | null | 438 | stdin | [
"4\n???\nY??D?X\n???\nD??DXYXYX\n"
] | [
"YES\nYDX\nX 0 D 0 Y 0 \nYES\nYDXDYX\nX 0 Y 0 D 1\nX 2 D 3 Y 4\nYES\nYDX\nY 0 D 1 X 2\nNO\n"
] | [
"4\n???\nY??D?X\n???\nD??DXYXYX\n",
"27\nYYY\nDYY\nXYY\nYDY\nDDY\nXDY\nYXY\nDXY\nXXY\nYYD\nDYD\nXYD\nYDD\nDDD\nXDD\nYXD\nDXD\nXXD\nYYX\nDYX\nXYX\nYDX\nDDX\nXDX\nYXX\nDXX\nXXX\n",
"64\nYYY\nDYY\nXYY\n?YY\nYDY\nDDY\nXDY\n?DY\nYXY\nDXY\nXXY\n?XY\nY?Y\nD?Y\nX?Y\n??Y\nYYD\nDYD\nXYD\n?YD\nYDD\nDDD\nXDD\n?DD\nYXD\nDXD\nXXD\n?XD\nY?D\nD?D\nX?D\n??D\nYYX\nDYX\nXYX\n?YX\nYDX\nDDX\nXDX\n?DX\nYXX\nDXX\nXXX\n?XX\nY?X\nD?X\nX?X\n??X\nYY?\nDY?\nXY?\n?Y?\nYD?\nDD?\nXD?\n?D?\nYX?\nDX?\nXX?\n?X?\nY??\nD??\nX??\n???\n"
] | [
"YES\nYDX\nX 0 D 0 Y 0 \nYES\nYDXDYX\nX 0 Y 0 D 0 \nX 0 D 0 Y 0 \nYES\nYDX\nX 0 D 0 Y 0 \nNO\n",
"NO\nNO\nNO\nNO\nNO\nYES\nXDY\nY 0 D 0 X 0 \nNO\nYES\nDXY\nY 0 X 0 D 0 \nNO\nNO\nNO\nYES\nXYD\nD 0 Y 0 X 0 \nNO\nNO\nNO\nYES\nYXD\nD 0 X 0 Y 0 \nNO\nNO\nNO\nYES\nDYX\nX 0 Y 0 D 0 \nNO\nYES\nYDX\nX 0 D 0 Y 0 \nNO\nNO\nNO\nNO\nNO\n",
"NO\nNO\nNO\nNO\nNO\nNO\nYES\nXDY\nX 0 D 1 Y 2\nYES\nXDY\nX 0 D 1 Y 2\nNO\nYES\nDXY\nD 0 X 1 Y 2\nNO\nYES\nDXY\nD 0 X 1 Y 2\nNO\nYES\nDXY\nD 0 X 1 Y 2\nYES\nXDY\nX 0 D 1 Y 2\nYES\nDXY\nD 0 X 1 Y 2\nNO\nNO\nYES\nXYD\nX 0 Y 1 D 2\nYES\nXYD\nX 0 Y 1 D 2\nNO\nNO\nNO\nNO\nYES\nYXD\nY 0 X 1 D 2\nNO\nNO\nYES\nYXD\nY 0 X 1 D 2\nYES\nYXD\nY 0 X 1 D 2\nNO\nYES\nXYD\nX 0 Y 1 D 2\nYES\nYXD\nY 0 X 1 D 2\nNO\nYES\nDYX\nD 0 Y 1 X 2\nNO\nYES\nDYX\nD 0 Y 1 X 2\nYES\nYDX\nY 0 D 1 X 2\nNO\nNO\nYES\nYDX\nY 0 D 1 X 2\nNO\nNO\nNO\nNO\nYES\nYDX\nY 0 D 1 X 2\nYES\nDYX\nD 0 Y 1 X 2\nNO\nYES\nYDX\nY 0 D 1 X 2\nNO\nYES\nDYX\nD 0 Y 1 X 2\nYES\nXYD\nX 0 Y 1 D 2\nYES\nXYD\nX 0 Y 1 D 2\nYES\nYDX\nY 0 D 1 X 2\nNO\nYES\nXDY\nX 0 D 1 Y 2\nYES\nXDY\nX 0 D 1 Y 2\nYES\nYXD\nY 0 X 1 D 2\nYES\nDXY\nD 0 X 1 Y 2\nNO\nYES\nDXY\nD 0 X 1 Y 2\nYES\nYDX\nY 0 D 1 X 2\nYES\nDYX\nD 0 Y 1 X 2\nYES\nXYD\nX 0 Y 1 D 2\nYES\nYDX\nY 0 D 1 X 2\n"
] | This is the hard version of the problem. The difference between the versions is that in this version, there is no restriction on the number of question marks. You can hack only if you solved all versions of this problem.
For a long time, no one could decipher Sumerian cuneiform. However, it has finally succumbed to pressure! Today, you have the chance to decipher Yandex cuneiform.
Yandex cuneiform is defined by the following rules:
1. An empty string is a Yandex cuneiform.
2. If you insert exactly one copy of each of the three letters 'Y', 'D', and 'X' into a Yandex cuneiform in such a way that no two adjacent letters become equal after the operation, you obtain a Yandex cuneiform.
3. If a string can't be obtained using the above rules, it is not a Yandex cuneiform.
You are given a template. A template is a string consisting of the characters 'Y', 'D', 'X', and '?'.
You need to check whether there exists a way to replace each question mark with 'Y', 'D', or 'X' to obtain a Yandex cuneiform, and if it exists, output any of the matching options, as well as a sequence of insertion operations to obtain the resulting cuneiform.
In this version of the problem, the number of question marks in the template can be arbitrary.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 5 \cdot 10^4$$$). The description of the test cases follows.
Each test case consists of a single line containing a template of length $$$n$$$ ($$$3 \leq n < 2 \cdot 10^5$$$, $$$n \bmod 3 = 0$$$), consisting only of characters 'Y', 'D', 'X', and '?'.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case, output a single line containing 'NO' if it is not possible to obtain a cuneiform from the given template.
Otherwise, output 'YES' on the first line, and on the second line, any obtainable cuneiform. After that, you need to output the sequence of operations that leads to the cuneiform you printed.
A sequence of operations is described by $$$\frac{n}{3}$$$ triples of pairs. A pair has the form c p, where $$$c$$$ is one of the letters 'Y', 'D', or 'X', and $$$p$$$ is the position at which the letter $$$c$$$ should be inserted. The insertion position is the number of letters to skip from the beginning of the string for the insertion. For example, after inserting the character 'D' into the string "YDX" with $$$p=3$$$, the result is "YDXD", and with $$$p=0$$$, it is "DYDX". Note that the index cannot exceed the current length of the string.
The operations are applied from top to bottom, left to right. After inserting each triple to the string, there should be no two adjacent identical characters.
Example Input 0:
4
???
Y??D?X
???
D??DXYXYX
Example Output 0:
YES
YDX
X 0 D 0 Y 0
YES
YDXDYX
X 0 Y 0 D 1
X 2 D 3 Y 4
YES
YDX
Y 0 D 1 X 2
NO
Notes:
In the second example, the string is transformed like this: $$$"" \to \mathtt{YDX} \to \mathtt{YDXDYX}$$$. | 1 |
Codeforces_test | null | 439 | stdin | [
"3\n5\n3 5 2 1 3\n2\nabfda\nafbfa\n2\n1 2\n3\nab\nabc\naa\n4\n5 -3 5 -3\n4\naaaa\nbcbc\naba\ncbcb\n"
] | [
"YES\nNO\nYES\nNO\nNO\nNO\nYES\nNO\nYES\n"
] | [
"3\n5\n3 5 2 1 3\n2\nabfda\nafbfa\n2\n1 2\n3\nab\nabc\naa\n4\n5 -3 5 -3\n4\naaaa\nbcbc\naba\ncbcb\n"
] | [
"YES\nNO\nYES\nNO\nNO\nNO\nYES\nNO\nYES\n"
] | Kristina has an array $$$a$$$, called a template, consisting of $$$n$$$ integers. She also has $$$m$$$ strings, each consisting only of lowercase Latin letters. The strings are numbered from $$$1$$$ to $$$m$$$. She wants to check which strings match the template.
A string $$$s$$$ is considered to match the template if all of the following conditions are simultaneously satisfied:
- The length of the string $$$s$$$ is equal to the number of elements in the array $$$a$$$.
- The same numbers from $$$a$$$ correspond to the same symbols from $$$s$$$. So, if $$$a_i = a_j$$$, then $$$s_i = s_j$$$ for ($$$1 \le i, j \le n$$$).
- The same symbols from $$$s$$$ correspond to the same numbers from $$$a$$$. So, if $$$s_i = s_j$$$, then $$$a_i = a_j$$$ for ($$$1 \le i, j \le n$$$).
For example, if $$$a$$$ = [$$$3, 5, 2, 1, 3$$$], then the string "abfda" matches the template, while the string "afbfa" does not, since the character "f" corresponds to both numbers $$$1$$$ and $$$5$$$.
Input format:
The first line of input contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.
The following descriptions are for the test cases.
The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of elements in the array $$$a$$$.
The second line of each test case contains exactly $$$n$$$ integers $$$a_i$$$ ($$$-10^9 \le a_i \le 10^9$$$) — the elements of the array $$$a$$$.
The third line of each test case contains a single integer $$$m$$$ ($$$1 \le m \le 2 \cdot 10^5$$$) — the number of strings to check for template matching.
Following are $$$m$$$ strings, each containing a non-empty string $$$s_j$$$ ($$$1 \le |s_j| \le 2 \cdot 10^5$$$), consisting of lowercase Latin letters.
It is guaranteed that the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$, and that the sum of the lengths of all strings does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case, output $$$m$$$ lines. On the $$$i$$$-th line ($$$1 \le i \le m$$$) output:
- "YES", if the string with index $$$i$$$ matches the template;
- "NO" otherwise.
You may output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as a positive answer).
Example Input 0:
3
5
3 5 2 1 3
2
abfda
afbfa
2
1 2
3
ab
abc
aa
4
5 -3 5 -3
4
aaaa
bcbc
aba
cbcb
Example Output 0:
YES
NO
YES
NO
NO
NO
YES
NO
YES
Notes:
The first test case is explained in the problem statement. | 1 |
Codeforces_test | null | 440 | stdin | [
"4\n1\n6\n3\n98\n"
] | [
"Kosuke\nSakurako\nKosuke\nSakurako\n"
] | [
"4\n1\n6\n3\n98\n",
"100\n100\n99\n98\n97\n96\n95\n94\n93\n92\n91\n90\n89\n88\n87\n86\n85\n84\n83\n82\n81\n80\n79\n78\n77\n76\n75\n74\n73\n72\n71\n70\n69\n68\n67\n66\n65\n64\n63\n62\n61\n60\n59\n58\n57\n56\n55\n54\n53\n52\n51\n50\n49\n48\n47\n46\n45\n44\n43\n42\n41\n40\n39\n38\n37\n36\n35\n34\n33\n32\n31\n30\n29\n28\n27\n26\n25\n24\n23\n22\n21\n20\n19\n18\n17\n16\n15\n14\n13\n12\n11\n10\n9\n8\n7\n6\n5\n4\n3\n2\n1\n"
] | [
"Kosuke\nSakurako\nKosuke\nSakurako\n",
"Sakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\nSakurako\nKosuke\n"
] | Sakurako and Kosuke decided to play some games with a dot on a coordinate line. The dot is currently located in position $$$x=0$$$. They will be taking turns, and Sakurako will be the one to start.
On the $$$i$$$-th move, the current player will move the dot in some direction by $$$2\cdot i-1$$$ units. Sakurako will always be moving the dot in the negative direction, whereas Kosuke will always move it in the positive direction.
In other words, the following will happen:
1. Sakurako will change the position of the dot by $$$-1$$$, $$$x = -1$$$ now
2. Kosuke will change the position of the dot by $$$3$$$, $$$x = 2$$$ now
3. Sakurako will change the position of the dot by $$$-5$$$, $$$x = -3$$$ now
4. $$$\cdots$$$
They will keep on playing while the absolute value of the coordinate of the dot does not exceed $$$n$$$. More formally, the game continues while $$$-n\le x\le n$$$. It can be proven that the game will always end.
Your task is to determine who will be the one who makes the last turn.
Input format:
The first line contains one integer $$$t$$$ ($$$1\le t\le 100$$$) — the number of games that Sakurako and Kosuke played.
Each game is described by one number $$$n$$$ ($$$1 \le n\le 100$$$) — the number that defines the condition when the game ends.
Output format:
For each of the $$$t$$$ games, output a line with the result of that game. If Sakurako makes the last turn, output "Sakurako" (without quotes); else output "Kosuke".
Example Input 0:
4
1
6
3
98
Example Output 0:
Kosuke
Sakurako
Kosuke
Sakurako
| 1 |
Codeforces_test | null | 441 | stdin | [
"3\n1\n0\n5\n00101\n00101\n11001\n00001\n11110\n6\n000000\n000000\n000000\n000000\n000000\n000000\n"
] | [
"1 \n4 2 1 3 5 \n6 5 4 3 2 1\n"
] | [
"3\n1\n0\n5\n00101\n00101\n11001\n00001\n11110\n6\n000000\n000000\n000000\n000000\n000000\n000000\n"
] | [
"1 \n4 2 1 3 5 \n6 5 4 3 2 1 \n"
] | You are given an undirected graph with $$$n$$$ vertices, labeled from $$$1$$$ to $$$n$$$. This graph encodes a hidden permutation$$$^{\text{∗}}$$$ $$$p$$$ of size $$$n$$$. The graph is constructed as follows:
- For every pair of integers $$$1 \le i < j \le n$$$, an undirected edge is added between vertex $$$p_i$$$ and vertex $$$p_j$$$ if and only if $$$p_i < p_j$$$. Note that the edge is not added between vertices $$$i$$$ and $$$j$$$, but between the vertices of their respective elements. Refer to the notes section for better understanding.
Your task is to reconstruct and output the permutation $$$p$$$. It can be proven that permutation $$$p$$$ can be uniquely determined.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 500$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$).
The $$$i$$$-th of the next $$$n$$$ lines contains a string of $$$n$$$ characters $$$g_{i, 1}g_{i, 2}\ldots g_{i, n}$$$ ($$$g_{i, j} = \mathtt{0}$$$ or $$$g_{i, j} = \mathtt{1}$$$) — the adjacency matrix. $$$g_{i, j} = \mathtt{1}$$$ if and only if there is an edge between vertex $$$i$$$ and vertex $$$j$$$.
It is guaranteed that there exists a permutation $$$p$$$ which generates the given graph. It is also guaranteed that the graph is undirected and has no self-loops, meaning $$$g_{i, j} = g_{j, i}$$$ and $$$g_{i, i} = \mathtt{0}$$$.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$1000$$$.
Output format:
For each test case, output $$$n$$$ integers $$$p_1, p_2, \ldots, p_n$$$ representing the reconstructed permutation.
Example Input 0:
3
1
0
5
00101
00101
11001
00001
11110
6
000000
000000
000000
000000
000000
000000
Example Output 0:
1
4 2 1 3 5
6 5 4 3 2 1
Notes:
In the first case $$$p = [1]$$$. Since there are no pairs $$$1 \le i < j \le n$$$, there are no edges in the graph.
The graph in the second case is shown below. For example, when we choose $$$i = 3$$$ and $$$j = 4$$$, we add an edge between vertices $$$p_i = 1$$$ and $$$p_j = 3$$$, because $$$p_i < p_j$$$. However, when we choose $$$i = 2$$$ and $$$j = 3$$$, $$$p_i = 2$$$ and $$$p_j = 1$$$, so $$$p_i < p_j$$$ doesn't hold. Therefore, we don't add an edge between $$$2$$$ and $$$1$$$.
In the third case, there are no edges in the graph, so there are no pairs of integers $$$1 \le i < j \le n$$$ such that $$$p_i < p_j$$$. Therefore, $$$p = [6, 5, 4, 3, 2, 1]$$$. | 1 |
Codeforces_test | null | 442 | stdin | [
"6\n1 1 3 5\n1 3 2 1\n8 10 28 100\n100 1 100 1\n1 100 1 100\n100 100 100 100\n"
] | [
"3\n2\n2\n1\n1\n2\n"
] | [
"6\n1 1 3 5\n1 3 2 1\n8 10 28 100\n100 1 100 1\n1 100 1 100\n100 100 100 100\n",
"1\n7 8 22 38\n"
] | [
"3\n2\n2\n1\n1\n2\n",
"1\n"
] | There is an array of $$$5$$$ integers. Initially, you only know $$$a_1,a_2,a_4,a_5$$$. You may set $$$a_3$$$ to any positive integer, negative integer, or zero. The Fibonacciness of the array is the number of integers $$$i$$$ ($$$1 \leq i \leq 3$$$) such that $$$a_{i+2}=a_i+a_{i+1}$$$. Find the maximum Fibonacciness over all integer values of $$$a_3$$$.
Input format:
The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 500$$$) — the number of test cases.
The only line of each test case contains four integers $$$a_1, a_2, a_4, a_5$$$ ($$$1 \leq a_i \leq 100$$$).
Output format:
For each test case, output the maximum Fibonacciness on a new line.
Example Input 0:
6
1 1 3 5
1 3 2 1
8 10 28 100
100 1 100 1
1 100 1 100
100 100 100 100
Example Output 0:
3
2
2
1
1
2
Notes:
In the first test case, we can set $$$a_3$$$ to $$$2$$$ to achieve the maximal Fibonacciness of $$$3$$$.
In the third test case, it can be shown that $$$2$$$ is the maximum Fibonacciness that can be achieved. This can be done by setting $$$a_3$$$ to $$$18$$$. | 1 |
Codeforces_test | null | 443 | stdin | [
"6\n1\nABCD\n2\nAAAAAAAA\n2\nAAAABBBB\n2\n????????\n3\nABCABCABCABC\n5\nACADC??ACAC?DCAABC?C\n"
] | [
"4\n2\n4\n0\n9\n13\n"
] | [
"6\n1\nABCD\n2\nAAAAAAAA\n2\nAAAABBBB\n2\n????????\n3\nABCABCABCABC\n5\nACADC??ACAC?DCAABC?C\n",
"1\n5\nA?B?C?D??D?C?B?A????\n"
] | [
"4\n2\n4\n0\n9\n13\n",
"8\n"
] | Tim is doing a test consisting of $$$4n$$$ questions; each question has $$$4$$$ options: 'A', 'B', 'C', and 'D'. For each option, there are exactly $$$n$$$ correct answers corresponding to that option — meaning there are $$$n$$$ questions with the answer 'A', $$$n$$$ questions with the answer 'B', $$$n$$$ questions with the answer 'C', and $$$n$$$ questions with the answer 'D'.
For each question, Tim wrote his answer on the answer sheet. If he could not figure out the answer, he would leave a question mark '?' for that question.
You are given his answer sheet of $$$4n$$$ characters. What is the maximum number of correct answers Tim can get?
Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases.
The first line of each test case contains an integer $$$n$$$ ($$$1 \le n \le 100$$$).
The second line of each test case contains a string $$$s$$$ of $$$4n$$$ characters ($$$s_i \in \{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}, \texttt{?}\}$$$) — Tim's answers for the questions.
Output format:
For each test case, print a single integer — the maximum score that Tim can achieve.
Example Input 0:
6
1
ABCD
2
AAAAAAAA
2
AAAABBBB
2
????????
3
ABCABCABCABC
5
ACADC??ACAC?DCAABC?C
Example Output 0:
4
2
4
0
9
13
Notes:
In the first test case, there is exactly one question with each answer 'A', 'B', 'C', and 'D'; so it's possible that Tim gets all his answers correct.
In the second test case, there are only two correct answers 'A' which makes him get exactly $$$2$$$ points in any case.
In the third test case, Tim can get at most $$$2$$$ correct answers with option 'A' and $$$2$$$ correct answers with option 'B'. For example, he would get $$$4$$$ points if the answers were 'AACCBBDD'.
In the fourth test case, he refuses to answer any question at all, which makes him get $$$0$$$ points. | 1 |
Codeforces_test | null | 444 | stdin | [
"3\n2\n8 0\n5\n2 2 2 2 2\n5\n0 0 9 0 0\n"
] | [
"8 4 \n4 5 4 5 4 \n4 6 9 6 4\n"
] | [
"3\n2\n8 0\n5\n2 2 2 2 2\n5\n0 0 9 0 0\n",
"2\n1\n1000000000\n4\n0 2 2 8\n"
] | [
"8 4 \n4 5 4 5 4 \n4 6 9 6 4 \n",
"1000000000 \n3 5 7 9 \n"
] | There are very long classes in the T-Generation. In one day, you need to have time to analyze the training and thematic contests, give a lecture with new material, and, if possible, also hold a mini-seminar. Therefore, there is a break where students can go to drink coffee and chat with each other.
There are a total of $$$n+2$$$ coffee machines located in sequentially arranged rooms along a long corridor. The coffee machines are numbered from $$$0$$$ to $$$n+1$$$, and immediately after the break starts, there are $$$a_i$$$ students gathered around the $$$i$$$-th coffee machine.
The students are talking too loudly among themselves, and the teachers need to make a very important announcement. Therefore, they want to gather the maximum number of students around some single coffee machine. The teachers are too lazy to run around the corridors and gather the students, so they came up with a more sophisticated way to manipulate them:
- At any moment, the teachers can choose room $$$i$$$ ($$$1 \le i \le n$$$) and turn off the lights there;
- If there were $$$x$$$ students in that room, then after turning off the lights, $$$\lfloor \frac12 x \rfloor$$$ students will go to room $$$(i-1)$$$, and $$$\lfloor \frac12 x \rfloor$$$ other students will go to room $$$(i+1)$$$.
- If $$$x$$$ was odd, then one student remains in the same room.
- After that, the lights in room $$$i$$$ are turned back on.
The teachers have not yet decided where they will gather the students, so for each $$$i$$$ from $$$1$$$ to $$$n$$$, you should determine what is the maximum number of students that can be gathered around the $$$i$$$-th coffee machine.
The teachers can turn off the lights in any rooms at their discretion, in any order, possibly turning off the lights in the same room multiple times.
Note that the values of $$$a_0$$$ and $$$a_{n+1}$$$ do not affect the answer to the problem, so their values will not be given to you.
Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10\,000$$$) — the number of test cases.
In the first line of each test case, there is an integer $$$n$$$ ($$$1 \le n \le 10^6$$$).
In the second line of each test case, there are integers $$$a_1, \ldots, a_n$$$ ($$$0 \le a_i \le 10^9$$$) — the number of students around coffee machines numbered $$$1, 2, \ldots, n$$$.
It is guaranteed that the sum of $$$n$$$ across all test cases does not exceed $$$10^6$$$.
Output format:
For each test case, output $$$n$$$ integers $$$b_1, \ldots, b_n$$$, where $$$b_i$$$ is the maximum number of students that can be around coffee machines number $$$i$$$.
Example Input 0:
3
2
8 0
5
2 2 2 2 2
5
0 0 9 0 0
Example Output 0:
8 4
4 5 4 5 4
4 6 9 6 4
Notes:
Let's analyze the first test case:
- To maximize the number of students at the $$$1$$$-st coffee machine, all that needs to be done is nothing.
- To maximize the number of students at the $$$2$$$-nd coffee machine, it is sufficient to turn off the lights in the $$$1$$$-st room once. | 1 |
Codeforces_test | null | 445 | stdin | [
"3\n2\n3\n6\n"
] | [
"uo\niae\noeiiua\n"
] | [
"3\n2\n3\n6\n",
"100\n39\n77\n67\n25\n81\n26\n50\n11\n73\n95\n86\n16\n90\n33\n14\n79\n12\n100\n68\n64\n60\n27\n41\n15\n34\n24\n3\n61\n83\n47\n57\n65\n99\n43\n40\n21\n94\n72\n82\n85\n23\n71\n76\n32\n10\n17\n30\n18\n44\n59\n35\n89\n6\n63\n7\n69\n62\n70\n4\n29\n92\n87\n31\n48\n36\n28\n45\n97\n93\n98\n56\n38\n58\n80\n8\n1\n74\n91\n53\n55\n54\n51\n96\n5\n42\n52\n9\n22\n78\n88\n75\n13\n66\n2\n37\n20\n49\n19\n84\n46\n",
"50\n68\n41\n92\n27\n1\n20\n34\n47\n3\n29\n43\n44\n72\n19\n48\n7\n8\n38\n91\n51\n36\n47\n91\n73\n97\n75\n34\n35\n87\n19\n31\n46\n62\n59\n92\n87\n40\n22\n57\n81\n84\n7\n3\n53\n65\n91\n47\n17\n49\n65\n",
"1\n53\n"
] | [
"ae\naei\naaeiou\n",
"aaaaaaaaeeeeeeeeoooooooouuuuuuuuiiiiiii\naaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeooooooooooooooouuuuuuuuuuuuuuuiiiiiiiiiiiiiii\naaaaaaaaaaaaaaeeeeeeeeeeeeeeooooooooooooouuuuuuuuuuuuuiiiiiiiiiiiii\naaaaaeeeeeooooouuuuuiiiii\naaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeoooooooooooooooouuuuuuuuuuuuuuuuiiiiiiiiiiiiiiii\naaaaaaeeeeeooooouuuuuiiiii\naaaaaaaaaaeeeeeeeeeeoooooooooouuuuuuuuuuiiiiiiiiii\naaaeeoouuii\naaaaaaaaaaaaaaaeeeeeeeeeeeeeeeooooooooooooooouuuuuuuuuuuuuuiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeooooooooooooooooooouuuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeooooooooooooooooouuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiii\naaaaeeeooouuuiii\naaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaeeeeeeeooooooouuuuuuiiiiii\naaaeeeooouuuii\naaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeoooooooooooooooouuuuuuuuuuuuuuuuiiiiiiiiiiiiiii\naaaeeeoouuii\naaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeeoooooooooooooooooooouuuuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaeeeeeeeeeeeeeeoooooooooooooouuuuuuuuuuuuuiiiiiiiiiiiii\naaaaaaaaaaaaaeeeeeeeeeeeeeooooooooooooouuuuuuuuuuuuuiiiiiiiiiiii\naaaaaaaaaaaaeeeeeeeeeeeeoooooooooooouuuuuuuuuuuuiiiiiiiiiiii\naaaaaaeeeeeeooooouuuuuiiiii\naaaaaaaaaeeeeeeeeoooooooouuuuuuuuiiiiiiii\naaaeeeooouuuiii\naaaaaaaeeeeeeeooooooouuuuuuuiiiiii\naaaaaeeeeeooooouuuuuiiii\naeo\naaaaaaaaaaaaaeeeeeeeeeeeeoooooooooooouuuuuuuuuuuuiiiiiiiiiiii\naaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeooooooooooooooooouuuuuuuuuuuuuuuuiiiiiiiiiiiiiiii\naaaaaaaaaaeeeeeeeeeeooooooooouuuuuuuuuiiiiiiiii\naaaaaaaaaaaaeeeeeeeeeeeeooooooooooouuuuuuuuuuuiiiiiiiiiii\naaaaaaaaaaaaaeeeeeeeeeeeeeooooooooooooouuuuuuuuuuuuuiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeeoooooooooooooooooooouuuuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiiii\naaaaaaaaaeeeeeeeeeooooooooouuuuuuuuiiiiiiii\naaaaaaaaeeeeeeeeoooooooouuuuuuuuiiiiiiii\naaaaaeeeeoooouuuuiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeooooooooooooooooooouuuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaeeeeeeeeeeeeeeeoooooooooooooouuuuuuuuuuuuuuiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeoooooooooooooooouuuuuuuuuuuuuuuuiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeooooooooooooooooouuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiii\naaaaaeeeeeooooouuuuiiii\naaaaaaaaaaaaaaaeeeeeeeeeeeeeeoooooooooooooouuuuuuuuuuuuuuiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeooooooooooooooouuuuuuuuuuuuuuuiiiiiiiiiiiiiii\naaaaaaaeeeeeeeoooooouuuuuuiiiiii\naaeeoouuii\naaaaeeeeooouuuiii\naaaaaaeeeeeeoooooouuuuuuiiiiii\naaaaeeeeoooouuuiii\naaaaaaaaaeeeeeeeeeooooooooouuuuuuuuuiiiiiiii\naaaaaaaaaaaaeeeeeeeeeeeeoooooooooooouuuuuuuuuuuuiiiiiiiiiii\naaaaaaaeeeeeeeooooooouuuuuuuiiiiiii\naaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiii\naaeoui\naaaaaaaaaaaaaeeeeeeeeeeeeeooooooooooooouuuuuuuuuuuuiiiiiiiiiiii\naaeeoui\naaaaaaaaaaaaaaeeeeeeeeeeeeeeoooooooooooooouuuuuuuuuuuuuuiiiiiiiiiiiii\naaaaaaaaaaaaaeeeeeeeeeeeeeoooooooooooouuuuuuuuuuuuiiiiiiiiiiii\naaaaaaaaaaaaaaeeeeeeeeeeeeeeoooooooooooooouuuuuuuuuuuuuuiiiiiiiiiiiiii\naeou\naaaaaaeeeeeeoooooouuuuuuiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeooooooooooooooooouuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiii\naaaaaaaeeeeeeoooooouuuuuuiiiiii\naaaaaaaaaaeeeeeeeeeeoooooooooouuuuuuuuuiiiiiiiii\naaaaaaaaeeeeeeeooooooouuuuuuuiiiiiii\naaaaaaeeeeeeoooooouuuuuiiiii\naaaaaaaaaeeeeeeeeeooooooooouuuuuuuuuiiiiiiiii\naaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeeooooooooooooooooooouuuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeooooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeeoooooooooooooooooooouuuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaeeeeeeeeeeeooooooooooouuuuuuuuuuuiiiiiiiiiii\naaaaaaaaeeeeeeeeoooooooouuuuuuuiiiiiii\naaaaaaaaaaaaeeeeeeeeeeeeoooooooooooouuuuuuuuuuuiiiiiiiiiii\naaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeoooooooooooooooouuuuuuuuuuuuuuuuiiiiiiiiiiiiiiii\naaeeooui\na\naaaaaaaaaaaaaaaeeeeeeeeeeeeeeeooooooooooooooouuuuuuuuuuuuuuuiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaaaaaeeeeeeeeeeeooooooooooouuuuuuuuuuiiiiiiiiii\naaaaaaaaaaaeeeeeeeeeeeooooooooooouuuuuuuuuuuiiiiiiiiiii\naaaaaaaaaaaeeeeeeeeeeeooooooooooouuuuuuuuuuuiiiiiiiiii\naaaaaaaaaaaeeeeeeeeeeoooooooooouuuuuuuuuuiiiiiiiiii\naaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeooooooooooooooooooouuuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiiii\naeoui\naaaaaaaaaeeeeeeeeeoooooooouuuuuuuuiiiiiiii\naaaaaaaaaaaeeeeeeeeeeeoooooooooouuuuuuuuuuiiiiiiiiii\naaeeoouui\naaaaaeeeeeoooouuuuiiii\naaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeoooooooooooooooouuuuuuuuuuuuuuuiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaeeeeeeeeeeeeeeeooooooooooooooouuuuuuuuuuuuuuuiiiiiiiiiiiiiii\naaaeeeooouuii\naaaaaaaaaaaaaaeeeeeeeeeeeeeooooooooooooouuuuuuuuuuuuuiiiiiiiiiiiii\nae\naaaaaaaaeeeeeeee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"aaaaaaaaaaaaaaeeeeeeeeeeeeeeoooooooooooooouuuuuuuuuuuuuiiiiiiiiiiiii\naaaaaaaaaeeeeeeeeoooooooouuuuuuuuiiiiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaeeeeeeooooouuuuuiiiii\na\naaaaeeeeoooouuuuiiii\naaaaaaaeeeeeeeooooooouuuuuuuiiiiii\naaaaaaaaaaeeeeeeeeeeooooooooouuuuuuuuuiiiiiiiii\naeo\naaaaaaeeeeeeoooooouuuuuuiiiii\naaaaaaaaaeeeeeeeeeooooooooouuuuuuuuiiiiiiii\naaaaaaaaaeeeeeeeeeooooooooouuuuuuuuuiiiiiiii\naaaaaaaaaaaaaaaeeeeeeeeeeeeeeeoooooooooooooouuuuuuuuuuuuuuiiiiiiiiiiiiii\naaaaeeeeoooouuuuiii\naaaaaaaaaaeeeeeeeeeeoooooooooouuuuuuuuuiiiiiiiii\naaeeoui\naaeeooui\naaaaaaaaeeeeeeeeoooooooouuuuuuuiiiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaaaaaeeeeeeeeeeoooooooooouuuuuuuuuuiiiiiiiiii\naaaaaaaaeeeeeeeooooooouuuuuuuiiiiiii\naaaaaaaaaaeeeeeeeeeeooooooooouuuuuuuuuiiiiiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaeeeeeeeeeeeeeeeooooooooooooooouuuuuuuuuuuuuuiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeeooooooooooooooooooouuuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaeeeeeeeeeeeeeeeooooooooooooooouuuuuuuuuuuuuuuiiiiiiiiiiiiiii\naaaaaaaeeeeeeeooooooouuuuuuuiiiiii\naaaaaaaeeeeeeeooooooouuuuuuuiiiiiii\naaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeooooooooooooooooouuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiii\naaaaeeeeoooouuuuiii\naaaaaaaeeeeeeoooooouuuuuuiiiiii\naaaaaaaaaaeeeeeeeeeooooooooouuuuuuuuuiiiiiiiii\naaaaaaaaaaaaaeeeeeeeeeeeeeoooooooooooouuuuuuuuuuuuiiiiiiiiiiii\naaaaaaaaaaaaeeeeeeeeeeeeoooooooooooouuuuuuuuuuuuiiiiiiiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeooooooooooooooooouuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiii\naaaaaaaaeeeeeeeeoooooooouuuuuuuuiiiiiiii\naaaaaeeeeeoooouuuuiiii\naaaaaaaaaaaaeeeeeeeeeeeeooooooooooouuuuuuuuuuuiiiiiiiiiii\naaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeoooooooooooooooouuuuuuuuuuuuuuuuiiiiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeooooooooooooooooouuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiii\naaeeoui\naeo\naaaaaaaaaaaeeeeeeeeeeeooooooooooouuuuuuuuuuiiiiiiiiii\naaaaaaaaaaaaaeeeeeeeeeeeeeooooooooooooouuuuuuuuuuuuuiiiiiiiiiiiii\naaaaaaaaaaaaaaaaaaaeeeeeeeeeeeeeeeeeeoooooooooooooooooouuuuuuuuuuuuuuuuuuiiiiiiiiiiiiiiiiii\naaaaaaaaaaeeeeeeeeeeooooooooouuuuuuuuuiiiiiiiii\naaaaeeeeooouuuiii\naaaaaaaaaaeeeeeeeeeeoooooooooouuuuuuuuuuiiiiiiiii\naaaaaaaaaaaaaeeeeeeeeeeeeeooooooooooooouuuuuuuuuuuuuiiiiiiiiiiiii\n",
"aaaaaaaaaaaeeeeeeeeeeeiiiiiiiiiiioooooooooouuuuuuuuuu\n"
] | Narek has to spend 2 hours with some 2-year-old kids at the kindergarten. He wants to teach them competitive programming, and their first lesson is about palindromes.
Narek found out that the kids only know the vowels of the English alphabet (the letters $$$\mathtt{a}$$$, $$$\mathtt{e}$$$, $$$\mathtt{i}$$$, $$$\mathtt{o}$$$, and $$$\mathtt{u}$$$), so Narek needs to make a string that consists of vowels only. After making the string, he'll ask the kids to count the number of subsequences that are palindromes. Narek wants to keep it simple, so he's looking for a string such that the amount of palindrome subsequences is minimal.
Help Narek find a string of length $$$n$$$, consisting of lowercase English vowels only (letters $$$\mathtt{a}$$$, $$$\mathtt{e}$$$, $$$\mathtt{i}$$$, $$$\mathtt{o}$$$, and $$$\mathtt{u}$$$), which minimizes the amount of palindrome$$$^{\dagger}$$$ subsequences$$$^{\ddagger}$$$ in it.
$$$^{\dagger}$$$ A string is called a palindrome if it reads the same from left to right and from right to left.
$$$^{\ddagger}$$$ String $$$t$$$ is a subsequence of string $$$s$$$ if $$$t$$$ can be obtained from $$$s$$$ by removing several (possibly, zero or all) characters from $$$s$$$ and concatenating the remaining ones, without changing their order. For example, $$$\mathtt{odocs}$$$ is a subsequence of $$$\texttt{c}{\color{red}{\texttt{od}}}\texttt{ef}{\color{red}{\texttt{o}}}\texttt{r}{\color{red}{\texttt{c}}}\texttt{e}{\color{red}{\texttt{s}}}$$$.
Input format:
The first line of the input contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. Subsequently, the description of each test case follows.
The only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the size of the string.
Output format:
For each test case, output any string of length $$$n$$$ that satisfies the above conditions.
Example Input 0:
3
2
3
6
Example Output 0:
uo
iae
oeiiua
Notes:
In the first example, $$$\texttt{uo}$$$ has only three palindrome subsequences: $$$\texttt{u}$$$, $$$\texttt{o}$$$, and the empty string. It can be shown that there is no better answer.
In the third example, $$$\texttt{oeiiua}$$$ has only eight palindrome subsequences: $$$\texttt{o}$$$, $$$\texttt{e}$$$, $$$\texttt{i}$$$, $$$\texttt{i}$$$, $$$\texttt{u}$$$, $$$\texttt{a}$$$, $$$\texttt{ii}$$$, and the empty string. It can be shown that there is no better answer. | 1 |
Codeforces_test | null | 446 | stdin | [
"2\n3 1\n2 1 2\n5 2\n2 1 4\n4 2 5\n"
] | [
"5\n15\n"
] | [
"2\n3 1\n2 1 2\n5 2\n2 1 4\n4 2 5\n"
] | [
"5\n15\n"
] | Claire loves drawing lines. She receives a sheet of paper with an $$$n \times n$$$ grid and begins drawing "lines" on it. Well—the concept of a "line" here is not what we usually think of. Claire refers each line to be a set of consecutive vertical grid cells. When she draws a line, these cells are all covered with black ink. Initially, all the cells are white, and drawing lines turns some of them black. After drawing a few lines, Claire wonders: how many ways she can color an additional white cell black so that the remaining white cells do not form a single connected component.
Two cells are directly connected if they share an edge. Two cells $$$x$$$ and $$$y$$$ are indirectly connected if there exists a sequence of cells $$$c_0, c_1, \ldots, c_k$$$ with $$$k > 1$$$ such that $$$c_0 = x$$$, $$$c_k = y$$$, and for every $$$i \in \{1, 2, \ldots, k\}$$$ the cells $$$c_i$$$ and $$$c_{i-1}$$$ are directly connected. A set of cells forms a single connected component if each pair of cells in the set is either directly or indirectly connected.
The grid has $$$n$$$ rows and $$$n$$$ columns, both indexed from $$$1$$$ to $$$n$$$. Claire will draw $$$q$$$ lines. The $$$i$$$-th line is drawn in the $$$y_i$$$-th column, from the $$$s_i$$$-th row to the $$$f_i$$$-th row, where $$$s_i \leq f_i$$$ for each $$$i \in \{1, 2, \ldots, q\}$$$. Note that the cells that are passed by at least one of the $$$q$$$ lines are colored black. The following figure shows an example of a $$$20\times 20$$$ grid with $$$q=67$$$ lines. The grid cells marked with red star symbols refer to the cells such that, if Claire colors that cell black, all white cells no longer form a single connected component.
You may assume that, after drawing the $$$q$$$ lines, the remaining white cells form a single connected component with at least three white cells.
Input format:
The first line contains exactly one integer $$$t$$$, indicating the number of test cases. For each test case, it begins with a line containing exactly two integers $$$n$$$ and $$$q$$$. This indicates that the grid is $$$n$$$ by $$$n$$$ and that Claire draws $$$q$$$ lines on it. Then $$$q$$$ lines follow. For each $$$i \in \{1, 2, \ldots, q\}$$$, the $$$i$$$-th line among the $$$q$$$ lines contains exactly three integers $$$y_i$$$, $$$s_i$$$, and $$$f_i$$$.
- $$$1\le t \le 125$$$
- $$$2\leq n \leq 10^9$$$
- $$$q\ge 1$$$; the sum of all $$$q$$$ values is at most $$$10^5$$$.
- $$$1\leq y_i \leq n$$$
- $$$1\leq s_i \leq f_i \leq n$$$
- There are at least three white cells and all white cells form a connected component.
Output format:
Print an integer on a line, indicating how many ways Claire can color an additional white cell black so that the remaining white cells do not form a single connected component.
Example Input 0:
2
3 1
2 1 2
5 2
2 1 4
4 2 5
Example Output 0:
5
15
| 1 |
Codeforces_test | null | 447 | stdin | [
"6\n7 2\n11 3\n55 13\n5801 6\n8919 64\n8765432 1\n"
] | [
"12\n18\n196\n1975581\n958900\n38416403456028\n"
] | [
"6\n7 2\n11 3\n55 13\n5801 6\n8919 64\n8765432 1\n",
"1\n2000000000 2000000000\n"
] | [
"12\n18\n196\n1975581\n958900\n38416403456028\n",
"0\n"
] | Iris looked at the stars and a beautiful problem emerged in her mind. She is inviting you to solve it so that a meteor shower is believed to form.
There are $$$n$$$ stars in the sky, arranged in a row. Iris has a telescope, which she uses to look at the stars.
Initially, Iris observes stars in the segment $$$[1, n]$$$, and she has a lucky value of $$$0$$$. Iris wants to look for the star in the middle position for each segment $$$[l, r]$$$ that she observes. So the following recursive procedure is used:
- First, she will calculate $$$m = \left\lfloor \frac{l+r}{2} \right\rfloor$$$.
- If the length of the segment (i.e. $$$r - l + 1$$$) is even, Iris will divide it into two equally long segments $$$[l, m]$$$ and $$$[m+1, r]$$$ for further observation.
- Otherwise, Iris will aim the telescope at star $$$m$$$, and her lucky value will increase by $$$m$$$; subsequently, if $$$l \neq r$$$, Iris will continue to observe two segments $$$[l, m-1]$$$ and $$$[m+1, r]$$$.
Iris is a bit lazy. She defines her laziness by an integer $$$k$$$: as the observation progresses, she will not continue to observe any segment $$$[l, r]$$$ with a length strictly less than $$$k$$$. In this case, please predict her final lucky value.
Input format:
Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^5$$$) — the number of test cases. The description of test cases follows.
The only line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \leq k \leq n \leq 2\cdot 10^9$$$).
Output format:
For each test case, output a single integer — the final lucky value.
Example Input 0:
6
7 2
11 3
55 13
5801 6
8919 64
8765432 1
Example Output 0:
12
18
196
1975581
958900
38416403456028
Notes:
In the first test case, at the beginning, Iris observes $$$[1, 7]$$$. Since $$$[1, 7]$$$ has an odd length, she aims at star $$$4$$$ and therefore increases her lucky value by $$$4$$$. Then it is split into $$$2$$$ new segments: $$$[1, 3]$$$ and $$$[5, 7]$$$. The segment $$$[1, 3]$$$ again has an odd length, so Iris aims at star $$$2$$$ and increases her lucky value by $$$2$$$. Then it is split into $$$2$$$ new segments: $$$[1, 1]$$$ and $$$[3, 3]$$$, both having a length less than $$$2$$$, so no further observation is conducted. For range $$$[5, 7]$$$, the progress is similar and the lucky value eventually increases by $$$6$$$. Therefore, the final lucky value is $$$4 + 2 + 6 = 12$$$.
In the last test case, Iris finally observes all the stars and the final lucky value is $$$1 + 2 + \cdots + 8\,765\,432 = 38\,416\,403\,456\,028$$$. | 1 |
Codeforces_test | null | 448 | stdin | [
"16\n51\n60\n61\n777\n12345689\n1000000000\n2002\n3001\n977\n989898986\n80\n800001\n96\n70\n15\n90\n"
] | [
"3\n2\n1\n0\n1\n3\n5\n4\n0\n7\n1\n2\n7\n0\n7\n3\n"
] | [
"16\n51\n60\n61\n777\n12345689\n1000000000\n2002\n3001\n977\n989898986\n80\n800001\n96\n70\n15\n90\n"
] | [
"3\n2\n1\n0\n1\n3\n5\n4\n0\n7\n1\n2\n7\n0\n7\n3\n"
] | You are given a positive integer $$$n$$$. In one operation, you can add to $$$n$$$ any positive integer whose decimal representation contains only the digit $$$9$$$, possibly repeated several times.
What is the minimum number of operations needed to make the number $$$n$$$ contain at least one digit $$$7$$$ in its decimal representation?
For example, if $$$n = 80$$$, it is sufficient to perform one operation: you can add $$$99$$$ to $$$n$$$, after the operation $$$n = 179$$$, which contains the digit $$$7$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The only line of each test case contains an integer $$$n$$$ ($$$10 \leq n \leq 10^9$$$).
Output format:
For each test case, output the minimum number of operations required for the number $$$n$$$ to contain the digit $$$7$$$.
Example Input 0:
16
51
60
61
777
12345689
1000000000
2002
3001
977
989898986
80
800001
96
70
15
90
Example Output 0:
3
2
1
0
1
3
5
4
0
7
1
2
7
0
7
3
Notes:
In the first test case, three operations are sufficient: $$$51 + 9 + 9 + 9 = 78$$$, which contains the digit $$$7$$$. It can be shown that it is impossible to achieve the goal in one or two operations.
In the second test case, two operations are sufficient: $$$60 + 9 + 9 = 78$$$.
In the third test case, one operation is sufficient: $$$61 + 9 = 70$$$.
In the fourth test case, $$$n$$$ already contains the digit $$$7$$$, so no operations are required.
In the fifth test case, you can add $$$99$$$ to $$$n$$$ to obtain a number containing the digit $$$7$$$. | 1 |
Codeforces_test | null | 449 | stdin | [
"3\n3\n2 5\n1 6\n3 4\n4\n1 6\n7 8\n2 3\n4 5\n6\n2 3\n1 6\n7 8\n9 12\n10 11\n4 5\n"
] | [
"5 1 1 1\n14 2 2 1 1\n132 42 5 2 1 1 1\n"
] | [
"3\n3\n2 5\n1 6\n3 4\n4\n1 6\n7 8\n2 3\n4 5\n6\n2 3\n1 6\n7 8\n9 12\n10 11\n4 5\n"
] | [
"5 1 1 1\n14 2 2 1 1\n132 42 5 2 1 1 1\n"
] | This is the hard version of the problem. The difference between the versions is that in this version, the limits on $$$t$$$ and $$$n$$$ are bigger. You can hack only if you solved all versions of this problem.
Now Little John is rich, and so he finally buys a house big enough to fit himself and his favorite bracket sequence. But somehow, he ended up with a lot of brackets! Frustrated, he penetrates through the ceiling with the "buddha palm".
A bracket sequence is called balanced if it can be constructed by the following formal grammar.
1. The empty sequence $$$\varnothing$$$ is balanced.
2. If the bracket sequence $$$A$$$ is balanced, then $$$\mathtt{(}A\mathtt{)}$$$ is also balanced.
3. If the bracket sequences $$$A$$$ and $$$B$$$ are balanced, then the concatenated sequence $$$A B$$$ is also balanced.
For example, the sequences "(())()", "()", "(()(()))", and the empty sequence are balanced, while "(()" and "(()))(" are not.
Given a balanced bracket sequence $$$s$$$, a pair of indices $$$(i,j)$$$ ($$$i<j$$$) is called a good pair if $$$s_i$$$ is '(', $$$s_j$$$ is ')', and the two brackets are added simultaneously with respect to Rule 2 while constructing the sequence $$$s$$$. For example, the sequence "(())()" has three different good pairs, which are $$$(1,4)$$$, $$$(2,3)$$$, and $$$(5,6)$$$. One can show that any balanced bracket sequence of $$$2n$$$ brackets contains exactly $$$n$$$ different good pairs, and using any order of rules to construct the same bracket sequence will yield the same set of good pairs.
Emily will play a bracket guessing game with John. The game is played as follows.
Initially, John has a balanced bracket sequence $$$s$$$ containing $$$n$$$ different good pairs, which is not known to Emily. John tells Emily the value of $$$n$$$ and asks Emily to guess the sequence.
Throughout $$$n$$$ turns, John gives Emily the following kind of clue on each turn.
- $$$l\;r$$$: The sequence $$$s$$$ contains a good pair $$$(l,r)$$$.
The clues that John gives Emily are pairwise distinct and do not contradict each other.
At a certain point, Emily can be certain that the balanced bracket sequence satisfying the clues given so far is unique. For example, assume Emily knows that $$$s$$$ has $$$3$$$ good pairs, and it contains the good pair $$$(2,5)$$$. Out of $$$5$$$ balanced bracket sequences with $$$3$$$ good pairs, there exists only one such sequence "((()))" with the good pair $$$(2,5)$$$. Therefore, one can see that Emily does not always need $$$n$$$ turns to guess $$$s$$$.
To find out the content of $$$s$$$ as early as possible, Emily wants to know the number of different balanced bracket sequences that match the clues after each turn. Surely, this is not an easy job for Emily, especially when she is given so many good pairs. Now it is your turn to help Emily. Given the clues, you must find the answer before and after each turn. As the answers may be huge, you need to find them modulo $$$998\,244\,353$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains one integer $$$n$$$ ($$$2 \le n \le 3 \cdot 10^5$$$) — the number of good pairs.
Then, each of the $$$n$$$ following lines contains two integers $$$l_i$$$ and $$$r_i$$$ representing the $$$i$$$-th clue ($$$1 \le l_i < r_i \le 2n$$$).
The clues in one test case are pairwise distinct and do not contradict each other.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.
Output format:
For each test case, output $$$n+1$$$ integers on a separate line:
- The first integer is the answer before all clues, modulo $$$998\,244\,353$$$.
- For all $$$i \ge 1$$$, the $$$i+1$$$-th integer is the answer after the $$$i$$$-th clue, modulo $$$998\,244\,353$$$.
Example Input 0:
3
3
2 5
1 6
3 4
4
1 6
7 8
2 3
4 5
6
2 3
1 6
7 8
9 12
10 11
4 5
Example Output 0:
5 1 1 1
14 2 2 1 1
132 42 5 2 1 1 1
Notes:
The first test case of the example is explained in the problem description.
The third test case of the example is explained as follows. It can be shown that there are $$$132$$$ balanced bracket sequences with $$$6$$$ good pairs. The answers after each clue are given as follows:
1. You are given the good pair $$$(2,3)$$$. There are $$$42$$$ balanced bracket sequences having the good pair $$$(2,3)$$$.
2. You are given the good pair $$$(1,6)$$$. There are $$$5$$$ balanced bracket sequences having good pairs $$$(2,3)$$$, $$$(1,6)$$$.
3. You are given the good pair $$$(7,8)$$$. There are $$$2$$$ balanced bracket sequences having the three good pairs. The strings are "(()())()(())" and "(()())()()()", respectively.
4. You are given the good pair $$$(9,12)$$$. There is only one balanced bracket sequence having the four good pairs. The content of $$$s$$$ is therefore the only string, which is "(()())()(())".
Then, the number of bracket sequences after the fifth and the sixth clue are both $$$1$$$ as you already know the content of $$$s$$$. | 1 |
Codeforces_test | null | 450 | stdin | [
"6\n2 1 1\n1 2\n4 1 2\n1 2\n2 4\n7 2 3\n1 2\n1 3\n6 7\n5 1 2\n1 2\n3 5\n9 2 1\n2 8\n9 2 4\n7 9\n4 8\n1 3\n2 3\n"
] | [
"1 1\n2 1\n1 4\n1 1\n1 1\n3 4\n"
] | [
"6\n2 1 1\n1 2\n4 1 2\n1 2\n2 4\n7 2 3\n1 2\n1 3\n6 7\n5 1 2\n1 2\n3 5\n9 2 1\n2 8\n9 2 4\n7 9\n4 8\n1 3\n2 3\n"
] | [
"1 1\n2 1\n1 4\n1 1\n1 1\n3 4\n"
] | Robin's brother and mother are visiting, and Robin gets to choose the start day for each visitor.
All days are numbered from $$$1$$$ to $$$n$$$. Visitors stay for $$$d$$$ continuous days, all of those $$$d$$$ days must be between day $$$1$$$ and $$$n$$$ inclusive.
Robin has a total of $$$k$$$ risky 'jobs' planned. The $$$i$$$-th job takes place between days $$$l_i$$$ and $$$r_i$$$ inclusive, for $$$1 \le i \le k$$$. If a job takes place on any of the $$$d$$$ days, the visit overlaps with this job (the length of overlap is unimportant).
Robin wants his brother's visit to overlap with the maximum number of distinct jobs, and his mother's the minimum.
Find suitable start days for the visits of Robin's brother and mother. If there are multiple suitable days, choose the earliest one.
Input format:
The first line of the input contains a single integer $$$t$$$ ($$$1\leq t \leq 10^4$$$) — the number of test cases.
The first line of each test case consists of three integers $$$n$$$, $$$d$$$, $$$k$$$ ($$$1 \le n \le 10^5, 1 \le d, k \le n$$$) — the number of total days, duration of the visits, and the number of jobs.
Then follow $$$k$$$ lines of each test case, each with two integers $$$l_i$$$ and $$$r_i$$$ ($$$1 \le l_i \le r_i \le n$$$) — the start and end day of each job.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case, output two integers, the best starting days of Robin's brother and mother respectively. Both visits must fit between day $$$1$$$ and $$$n$$$ inclusive.
Example Input 0:
6
2 1 1
1 2
4 1 2
1 2
2 4
7 2 3
1 2
1 3
6 7
5 1 2
1 2
3 5
9 2 1
2 8
9 2 4
7 9
4 8
1 3
2 3
Example Output 0:
1 1
2 1
1 4
1 1
1 1
3 4
Notes:
In the first test case, the only job fills all $$$2$$$ days, both should visit on day $$$1$$$.
In the second test case, day $$$2$$$ overlaps with $$$2$$$ jobs and day $$$1$$$ overlaps with only $$$1$$$.
In the third test case, Robert visits for days $$$[1,2]$$$, Mrs. Hood visits for days $$$[4,5]$$$. | 1 |
Codeforces_test | null | 451 | stdin | [
"2\n4 3\n1 5 2 4\n4 9 5 3\n4 5 2 3\n1 5 5 2\n1 1 4 4\n2 2 3 3\n1 2 4 3\n3 3\n1 2 3\n4 5 6\n7 8 9\n1 1 1 3\n1 3 3 3\n2 2 2 2\n"
] | [
"500 42 168 \n14 42 5\n"
] | [
"2\n4 3\n1 5 2 4\n4 9 5 3\n4 5 2 3\n1 5 5 2\n1 1 4 4\n2 2 3 3\n1 2 4 3\n3 3\n1 2 3\n4 5 6\n7 8 9\n1 1 1 3\n1 3 3 3\n2 2 2 2\n"
] | [
"500 42 168 \n14 42 5 \n"
] | Swing is opening a pancake factory! A good pancake factory must be good at flattening things, so Swing is going to test his new equipment on 2D matrices.
Swing is given an $$$n \times n$$$ matrix $$$M$$$ containing positive integers. He has $$$q$$$ queries to ask you.
For each query, he gives you four integers $$$x_1$$$, $$$y_1$$$, $$$x_2$$$, $$$y_2$$$ and asks you to flatten the submatrix bounded by $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ into an array $$$A$$$. Formally, $$$A = [M_{(x1,y1)}, M_{(x1,y1+1)}, \ldots, M_{(x1,y2)}, M_{(x1+1,y1)}, M_{(x1+1,y1+1)}, \ldots, M_{(x2,y2)}]$$$.
The following image depicts the flattening of a submatrix bounded by the red dotted lines. The orange arrows denote the direction that the elements of the submatrix are appended to the back of $$$A$$$, and $$$A$$$ is shown at the bottom of the image.
Afterwards, he asks you for the value of $$$\sum_{i=1}^{|A|} A_i \cdot i$$$ (sum of $$$A_i \cdot i$$$ over all $$$i$$$).
Input format:
The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases.
The first line of each test contains two integers $$$n$$$ and $$$q$$$ ($$$1 \leq n \leq 2000, 1 \leq q \leq 10^6$$$) — the length of $$$M$$$ and the number of queries.
The following $$$n$$$ lines contain $$$n$$$ integers each, the $$$i$$$'th of which contains $$$M_{(i,1)}, M_{(i,2)}, \ldots, M_{(i,n)}$$$ ($$$1 \leq M_{(i, j)} \leq 10^6$$$).
The following $$$q$$$ lines contain four integers $$$x_1$$$, $$$y_1$$$, $$$x_2$$$, and $$$y_2$$$ ($$$1 \leq x_1 \leq x_2 \leq n, 1 \leq y_1 \leq y_2 \leq n$$$) — the bounds of the query.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2000$$$ and the sum of $$$q$$$ over all test cases does not exceed $$$10^6$$$.
Output format:
For each test case, output the results of the $$$q$$$ queries on a new line.
Example Input 0:
2
4 3
1 5 2 4
4 9 5 3
4 5 2 3
1 5 5 2
1 1 4 4
2 2 3 3
1 2 4 3
3 3
1 2 3
4 5 6
7 8 9
1 1 1 3
1 3 3 3
2 2 2 2
Example Output 0:
500 42 168
14 42 5
Notes:
In the second query of the first test case, $$$A = [9, 5, 5, 2]$$$. Therefore, the sum is $$$1 \cdot 9 + 2 \cdot 5 + 3 \cdot 5 + 4 \cdot 2 = 42$$$. | 1 |
Codeforces_test | null | 452 | stdin | [
"6\n2 3\n10\n2 3\n11\n5 1\n11101\n7 9\n1101011\n17 34\n11001010001010010\n1 500\n1\n"
] | [
"3\n2\n0\n8\n32\n0\n"
] | [
"6\n2 3\n10\n2 3\n11\n5 1\n11101\n7 9\n1101011\n17 34\n11001010001010010\n1 500\n1\n",
"4\n5 347\n11111\n61 373\n1000000000000000000000000000100000000000000000000000000000000\n57 439\n100010101011110000010111010000110010000110101000110111010\n77 488\n10000000000000000000000000000000000010000000000000000000000000000000100000000\n",
"4\n43 136\n1111100010001010100011011110110001110100001\n10 20\n1000101110\n17 6\n10100000100100110\n130 68\n1011111111111111111111111111011111111111111111110111111111111111111111111111111111111011111111011111111111111111111111110111111111\n",
"4\n45 12\n111010001111111001110111111111011011101100111\n24 38\n101000001101111110011001\n19 55\n1000001110111110010\n112 40\n1100000001110100001001000000100000110010000000010000000000000000000000000010100000000001000000000010000000000000\n",
"4\n69 3\n110111011100000110000010011111101100000100010000100100000100011000110\n1 1\n1\n4 1\n1000\n126 1\n110101110011000010110101101101011111110001000110001111010011000101101011101110001111111011110001000111111111001100101011010101\n",
"4\n44 331\n11111111111111111111111111111111111111111111\n99 395\n111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n1 488\n1\n56 493\n11111111111111111111111111111111111111111111111111111111\n",
"4\n119 13\n11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n39 17\n111111111111111111111111111111111111111\n13 13\n1111111111111\n29 6\n11111111111111111111111111111\n",
"4\n65 131\n11111111111111111111111111111111111111111111111111111111111111111\n52 326\n1111111111111111111111111111111111111111111111111111\n41 73\n11111111111111111111111111111111111111111\n42 332\n111111111111111111111111111111111111111111\n",
"4\n6 176\n111111\n54 140\n111111111111111111111111111111111111111111111111111111\n131 267\n11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n9 58\n111111111\n",
"4\n82 388\n1111111111111111111111111111111111111111111111111111111111111111111111111111111111\n56 200\n11111111111111111111111111111111111111111111111111111111\n57 75\n111111111111111111111111111111111111111111111111111111111\n5 21\n11111\n",
"1\n200 268\n11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n",
"1\n200 265\n11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n",
"4\n44 320\n11111111111111111111111111111111111111111111\n99 128\n111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n1 490\n1\n56 320\n11111111111111111111111111111111111111111111111111111111\n",
"4\n60 2\n111111111111111111111111111111111111111111111111111111111111\n32 386\n11111111111111111111111111111111\n64 395\n1111111111111111111111111111111111111111111111111111111111111111\n16 353\n1111111111111111\n",
"4\n7 33\n1111111\n64 419\n1111111111111111111111111111111111111111111111111111111111111111\n64 421\n1111111111111111111111111111111111111111111111111111111111111111\n32 199\n11111111111111111111111111111111\n",
"4\n32 65\n11111111111111111111111111111111\n32 217\n11111111111111111111111111111111\n32 258\n11111111111111111111111111111111\n64 1\n1111111111111111111111111111111111111111111111111111111111111111\n",
"1\n192 335\n111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n",
"1\n192 308\n111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n",
"4\n124 115\n1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n40 111\n1111111111111111111111111111111111111111\n33 60\n111111111111111111111111111111111\n3 264\n111\n",
"4\n94 379\n1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111\n3 108\n111\n31 128\n1111111111111111111111111111111\n72 466\n111111111111111111111111111111111111111111111111111111111111111111111111\n"
] | [
"3\n2\n0\n8\n32\n0\n",
"344\n374\n0\n0\n",
"6\n3\n1\n77\n",
"3\n38\n49\n0\n",
"2\n0\n0\n0\n",
"323\n3\n0\n486\n",
"12\n3\n12\n6\n",
"128\n5\n0\n333\n",
"0\n4\n3\n57\n",
"385\n219\n3\n3\n",
"10\n",
"11\n",
"0\n0\n1\n24\n",
"0\n384\n449\n368\n",
"32\n461\n458\n227\n",
"96\n236\n256\n0\n",
"471\n",
"473\n",
"14\n24\n3\n0\n",
"6\n3\n0\n31\n"
] | This is the easy version of the problem. The difference between the versions is that in this version, the constraints on $$$t$$$, $$$k$$$, and $$$m$$$ are lower. You can hack only if you solved all versions of this problem.
A sequence $$$a$$$ of $$$n$$$ integers is called good if the following condition holds:
- Let $$$\text{cnt}_x$$$ be the number of occurrences of $$$x$$$ in sequence $$$a$$$. For all pairs $$$0 \le i < j < m$$$, at least one of the following has to be true: $$$\text{cnt}_i = 0$$$, $$$\text{cnt}_j = 0$$$, or $$$\text{cnt}_i \le \text{cnt}_j$$$. In other words, if both $$$i$$$ and $$$j$$$ are present in sequence $$$a$$$, then the number of occurrences of $$$i$$$ in $$$a$$$ is less than or equal to the number of occurrences of $$$j$$$ in $$$a$$$.
You are given integers $$$n$$$ and $$$m$$$. Calculate the value of the bitwise XOR of the median$$$^{\text{∗}}$$$ of all good sequences $$$a$$$ of length $$$n$$$ with $$$0\le a_i < m$$$.
Note that the value of $$$n$$$ can be very large, so you are given its binary representation instead.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 50$$$). The description of the test cases follows.
The first line of each test case contains two integers $$$k$$$ and $$$m$$$ ($$$1 \le k \le 200$$$, $$$1 \le m \le 500$$$) — the number of bits in $$$n$$$ and the upper bound on the elements in sequence $$$a$$$.
The second line of each test case contains a binary string of length $$$k$$$ — the binary representation of $$$n$$$ with no leading zeros.
It is guaranteed that the sum of $$$k$$$ over all test cases does not exceed $$$200$$$.
Output format:
For each test case, output a single integer representing the bitwise XOR of the median of all good sequences $$$a$$$ of length $$$n$$$ where $$$0\le a_i < m$$$.
Example Input 0:
6
2 3
10
2 3
11
5 1
11101
7 9
1101011
17 34
11001010001010010
1 500
1
Example Output 0:
3
2
0
8
32
0
Notes:
In the first example, $$$n = 10_2 = 2$$$ and $$$m = 3$$$. All possible sequences with elements less than $$$m$$$ are: $$$[0, 0]$$$, $$$[0, 1]$$$, $$$[0, 2]$$$, $$$[1, 0]$$$, $$$[1, 1]$$$, $$$[1, 2]$$$, $$$[2, 0]$$$, $$$[2, 1]$$$, $$$[2, 2]$$$. All of them are good, so the answer is: $$$0 \oplus 0 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 0 \oplus 1 \oplus 2 = 3$$$.
In the second example, $$$n = 11_2 = 3$$$ and $$$m = 3$$$. Some good sequences are $$$[2, 2, 2]$$$, $$$[1, 0, 1]$$$, and $$$[2, 0, 1]$$$. However, a sequence $$$[2, 0, 0]$$$ is not good, because $$$\text{cnt}_0 = 2$$$, $$$\text{cnt}_2 = 1$$$. Therefore, if we set $$$i = 0$$$ and $$$j = 2$$$, $$$i < j$$$ holds, but $$$\text{cnt}_i \le \text{cnt}_j$$$ does not. | 1 |
Codeforces_test | null | 453 | stdin | [
"3\n3\n1 2\n2 3\n2 2\n3\n1 2\n2 3\n3 3\n6\n1 2\n1 3\n2 4\n2 5\n1 6\n4 4\n"
] | [
"Bob\nAlice\nAlice\n"
] | [
"3\n3\n1 2\n2 3\n2 2\n3\n1 2\n2 3\n3 3\n6\n1 2\n1 3\n2 4\n2 5\n1 6\n4 4\n"
] | [
"Bob\nAlice\nAlice\n"
] | This is the easy version of the problem. In this version, $$$\mathbf{u = v}$$$. You can make hacks only if both versions of the problem are solved.
Alice and Bob are playing a fun game on a tree. This game is played on a tree with $$$n$$$ vertices, numbered from $$$1$$$ to $$$n$$$. Recall that a tree with $$$n$$$ vertices is an undirected connected graph with $$$n - 1$$$ edges.
Alice and Bob take turns, with Alice going first. Each player starts at some vertex.
On their turn, a player must move from the current vertex to a neighboring vertex that has not yet been visited by anyone. The first player who cannot make a move loses.
You are given two vertices $$$u$$$ and $$$v$$$. Represent the simple path from vertex $$$u$$$ to $$$v$$$ as an array $$$p_1, p_2, p_3, \ldots, p_m$$$, where $$$p_1 = u$$$, $$$p_m = v$$$, and there is an edge between $$$p_i$$$ and $$$p_{i + 1}$$$ for all $$$i$$$ ($$$1 \le i < m$$$).
You need to determine the winner of the game if Alice starts at vertex $$$1$$$ and Bob starts at vertex $$$p_j$$$ for each $$$j$$$ (where $$$1 \le j \le m$$$).
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 2 \cdot 10^5$$$) — the number of vertices in the tree.
Each of the following $$$n - 1$$$ lines contains two integers $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le n$$$), denoting an undirected edge between vertices $$$a$$$ and $$$b$$$. It is guaranteed that these edges form a tree.
The last line of each test case contains two integers $$$u$$$ and $$$v$$$ ($$$2 \le u, v \le n$$$, $$$\mathbf{u = v}$$$).
It is guaranteed that the path from $$$u$$$ to $$$v$$$ does not pass through vertex $$$1$$$.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case, output $$$m$$$ lines.
In the $$$i$$$-th line, print the winner of the game if Alice starts at vertex $$$1$$$ and Bob starts at vertex $$$p_i$$$. Print "Alice" (without quotes) if Alice wins, or "Bob" (without quotes) otherwise.
Example Input 0:
3
3
1 2
2 3
2 2
3
1 2
2 3
3 3
6
1 2
1 3
2 4
2 5
1 6
4 4
Example Output 0:
Bob
Alice
Alice
Notes:
Tree from the first and second examples.
In the first test case, the path will be ($$$2,2$$$). Bob starts at vertex $$$2$$$, Alice will not be able to move anywhere on her first turn and will lose.
In the second test case, the path will be ($$$3,3$$$). Bob starts at vertex $$$3$$$, Alice will move to vertex $$$2$$$, and Bob will have no remaining vertices to visit and will lose. | 1 |
Codeforces_test | null | 454 | stdin | [
"9 5\n0 1 2 1 3 4 5 6 0\n1 5\n2 5\n3 5\n4 5\n1 9\n"
] | [
"4 1\n2 1\n0 1\n-1\n8 2\n"
] | [
"9 5\n0 1 2 1 3 4 5 6 0\n1 5\n2 5\n3 5\n4 5\n1 9\n",
"1 1\n0\n1 1\n",
"1 1\n50\n1 1\n"
] | [
"4 1\n2 1\n0 1\n-1\n8 2\n",
"0 1\n",
"-1\n"
] | Recall the rules of the game "Nim". There are $$$n$$$ piles of stones, where the $$$i$$$-th pile initially contains some number of stones. Two players take turns choosing a non-empty pile and removing any positive (strictly greater than $$$0$$$) number of stones from it. The player unable to make a move loses the game.
You are given an array $$$a$$$, consisting of $$$n$$$ integers. Artem and Ruslan decided to play Nim on segments of this array. Each of the $$$q$$$ rounds is defined by a segment $$$(l_i, r_i)$$$, where the elements $$$a_{l_i}, a_{l_i+1}, \dots, a_{r_i}$$$ represent the sizes of the piles of stones.
Before the game starts, Ruslan can remove any number of piles from the chosen segment. However, at least one pile must remain, so in a single round he can remove at most $$$(r_i - l_i)$$$ piles. He is allowed to remove $$$0$$$ piles. After the removal, the game is played on the remaining piles within the segment.
All rounds are independent: the changes made in one round do not affect the original array or any other rounds.
Ruslan wants to remove as many piles as possible so that Artem, who always makes the first move, loses.
For each round, determine:
1. the maximum number of piles Ruslan can remove;
2. the number of ways to choose the maximum number of piles for removal.
Two ways are considered different if there exists an index $$$i$$$ such that the pile at index $$$i$$$ is removed in one way but not in the other. Since the number of ways can be large, output it modulo $$$998\,244\,353$$$.
If Ruslan cannot ensure Artem's loss in a particular round, output -1 for that round.
Input format:
The first line of input contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 10^5$$$) — the size of the array and the number of segments for which the answers need to be calculated.
The second line of input contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$0 \le a_i \le 50$$$) — the elements of the initial array.
The $$$i$$$-th of the next $$$q$$$ lines contains two integers $$$l_i, r_i$$$ ($$$1 \le l_i \le r_i \le n$$$) — the bounds of the segment on which the boys want to play the game during the $$$i$$$-th round.
Output format:
For each round:
- if Ruslan can win, print two integers — the maximum number of piles that can be removed, and the number of ways to remove the maximum number of piles, taken modulo $$$998\,244\,353$$$;
- otherwise print -1.
Example Input 0:
9 5
0 1 2 1 3 4 5 6 0
1 5
2 5
3 5
4 5
1 9
Example Output 0:
4 1
2 1
0 1
-1
8 2
| 1 |
Codeforces_test | null | 455 | stdin | [
"4\n4\n2 0\n1 1\n3 0\n5 1\n6\n4 6\n1 3\n4 6\n4 0\n7 6\n6 3\n7\n9 0\n7 1\n5 0\n7 1\n9 0\n1 1\n2 0\n10\n10 7\n4 9\n2 2\n7 9\n2 8\n8 5\n11 7\n15 5\n12 7\n4 0\n"
] | [
"2 2 4 5 \n4 4 7 7 10 10 \n9 9 9 9 9 9 10 \n10 10 10 10 10 10 12 15 15 15\n"
] | [
"4\n4\n2 0\n1 1\n3 0\n5 1\n6\n4 6\n1 3\n4 6\n4 0\n7 6\n6 3\n7\n9 0\n7 1\n5 0\n7 1\n9 0\n1 1\n2 0\n10\n10 7\n4 9\n2 2\n7 9\n2 8\n8 5\n11 7\n15 5\n12 7\n4 0\n",
"1\n10\n1000000000 1\n1 0\n1000000000 1\n1 0\n1000000000 1\n1 0\n1000000000 1\n1 0\n1000000000 1\n1 0\n"
] | [
"2 2 4 5 \n4 4 7 7 10 10 \n9 9 9 9 9 9 10 \n10 10 10 10 10 10 12 15 15 15 \n",
"1000000000 1000000000 1999999999 1999999999 2999999998 2999999998 3999999997 3999999997 4999999996 4999999996 \n"
] | Given an array of integers $$$s_1, s_2, \ldots, s_l$$$, every second, cosmic rays will cause all $$$s_i$$$ such that $$$i=1$$$ or $$$s_i\neq s_{i-1}$$$ to be deleted simultaneously, and the remaining parts will be concatenated together in order to form the new array $$$s_1, s_2, \ldots, s_{l'}$$$.
Define the strength of an array as the number of seconds it takes to become empty.
You are given an array of integers compressed in the form of $$$n$$$ pairs that describe the array left to right. Each pair $$$(a_i,b_i)$$$ represents $$$a_i$$$ copies of $$$b_i$$$, i.e. $$$\underbrace{b_i,b_i,\cdots,b_i}_{a_i\textrm{ times}}$$$.
For each $$$i=1,2,\dots,n$$$, please find the strength of the sequence described by the first $$$i$$$ pairs.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1\le t\le10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1\le n\le3\cdot10^5$$$) — the length of sequence $$$a$$$.
The next $$$n$$$ lines contain two integers each $$$a_i$$$, $$$b_i$$$ ($$$1\le a_i\le10^9,0\le b_i\le n$$$) — the pairs which describe the sequence.
It is guaranteed that the sum of all $$$n$$$ does not exceed $$$3\cdot10^5$$$.
It is guaranteed that for all $$$1\le i<n$$$, $$$b_i\neq b_{i+1}$$$ holds.
Output format:
For each test case, print one line containing $$$n$$$ integers — the answer for each prefix of pairs.
Example Input 0:
4
4
2 0
1 1
3 0
5 1
6
4 6
1 3
4 6
4 0
7 6
6 3
7
9 0
7 1
5 0
7 1
9 0
1 1
2 0
10
10 7
4 9
2 2
7 9
2 8
8 5
11 7
15 5
12 7
4 0
Example Output 0:
2 2 4 5
4 4 7 7 10 10
9 9 9 9 9 9 10
10 10 10 10 10 10 12 15 15 15
Notes:
In the first test case, for the prefix of length $$$4$$$, the changes will be $$$[0,0,1,0,0,0,1,1,1,1,1]\rightarrow[0,0,0,1,1,1,1]\rightarrow[0,0,1,1,1]\rightarrow[0,1,1]\rightarrow[1]\rightarrow[]$$$, so the array becomes empty after $$$5$$$ seconds.
In the second test case, for the prefix of length $$$4$$$, the changes will be $$$[6,6,6,6,3,6,6,6,6,0,0,0,0]\rightarrow[6,6,6,6,6,6,0,0,0]\rightarrow[6,6,6,6,6,0,0]\rightarrow[6,6,6,6,0]\rightarrow[6,6,6]\rightarrow[6,6]\rightarrow[6]\rightarrow[]$$$, so the array becomes empty after $$$7$$$ seconds. | 1 |
Codeforces_test | null | 456 | stdin | [
"7\n4\n5 5 5 10\n4\n10 5 10 5\n4\n1 2 3 4\n4\n1 1 1 3\n6\n4 2 1 5 7 1\n6\n10 200 30 300 30 100\n4\n100000000 100000000 1 2\n"
] | [
"5 5 5 10\n5 5 10 10\n-1\n-1\n1 1 4 5\n-1\n100000000 100000000 1 2\n"
] | [
"7\n4\n5 5 5 10\n4\n10 5 10 5\n4\n1 2 3 4\n4\n1 1 1 3\n6\n4 2 1 5 7 1\n6\n10 200 30 300 30 100\n4\n100000000 100000000 1 2\n",
"1\n4\n1 2 2 4\n",
"1\n5\n8 10 10 12 100\n",
"1\n4\n10 11 11 12\n",
"1\n4\n100 105 105 106\n",
"1\n4\n1 5 5 6\n",
"1\n4\n1 10 10 20\n",
"1\n4\n3 4 4 5\n",
"1\n4\n1 2 2 3\n",
"1\n4\n1 1 1 1\n",
"1\n4\n4 4 4 4\n"
] | [
"5 10 5 5\n5 5 10 10\n-1\n-1\n4 5 1 1\n-1\n1 2 100000000 100000000\n",
"1 4 2 2\n",
"8 12 10 10\n",
"10 12 11 11\n",
"100 106 105 105\n",
"1 6 5 5\n",
"1 20 10 10\n",
"3 5 4 4\n",
"1 3 2 2\n",
"1 1 1 1\n",
"4 4 4 4\n"
] | Kevin has $$$n$$$ sticks with length $$$a_1,a_2,\ldots,a_n$$$.
Kevin wants to select $$$4$$$ sticks from these to form an isosceles trapezoid$$$^{\text{∗}}$$$ with a positive area. Note that rectangles and squares are also considered isosceles trapezoids. Help Kevin find a solution. If no solution exists, output $$$-1$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$4 \le n \le 2\cdot 10^5$$$).
The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^8$$$).
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2\cdot 10^5$$$.
Output format:
For each test case, output $$$4$$$ integers — the lengths of sticks. If no solution exists, output $$$-1$$$.
If there are multiple solutions, print any of them.
Example Input 0:
7
4
5 5 5 10
4
10 5 10 5
4
1 2 3 4
4
1 1 1 3
6
4 2 1 5 7 1
6
10 200 30 300 30 100
4
100000000 100000000 1 2
Example Output 0:
5 5 5 10
5 5 10 10
-1
-1
1 1 4 5
-1
100000000 100000000 1 2
Notes:
In the first test case, you can form an isosceles trapezoid with bases of length $$$5$$$ and $$$10$$$, and two legs of length $$$5$$$.
In the second test case, you can form an isosceles trapezoid with two bases of length $$$5$$$ and two legs of length $$$10$$$. A rectangle is considered an isosceles trapezoid here.
In the third test case, there are no sticks with the same length. It's impossible to form an isosceles trapezoid.
In the fourth test case, it's impossible to form an isosceles trapezoid with a positive area. | 1 |
Codeforces_test | null | 457 | stdin | [
"7\n1\n1\n3\n1 2 3\n3\n1 2 2\n5\n5 4 3 2 1\n6\n1 1 2 2 3 3\n8\n8 7 6 3 8 7 6 3\n6\n1 1 4 5 1 4\n"
] | [
"0\n2\n1\n4\n4\n6\n3\n"
] | [
"7\n1\n1\n3\n1 2 3\n3\n1 2 2\n5\n5 4 3 2 1\n6\n1 1 2 2 3 3\n8\n8 7 6 3 8 7 6 3\n6\n1 1 4 5 1 4\n"
] | [
"0\n2\n1\n4\n4\n6\n3\n"
] | You are given a cyclic array $$$a_1, a_2, \ldots, a_n$$$.
You can perform the following operation on $$$a$$$ at most $$$n - 1$$$ times:
- Let $$$m$$$ be the current size of $$$a$$$, you can choose any two adjacent elements where the previous one is no greater than the latter one (In particular, $$$a_m$$$ and $$$a_1$$$ are adjacent and $$$a_m$$$ is the previous one), and delete exactly one of them. In other words, choose an integer $$$i$$$ ($$$1 \leq i \leq m$$$) where $$$a_i \leq a_{(i \bmod m) + 1}$$$ holds, and delete exactly one of $$$a_i$$$ or $$$a_{(i \bmod m) + 1}$$$ from $$$a$$$.
Your goal is to find the minimum number of operations needed to make all elements in $$$a$$$ equal.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 500$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the length of the array $$$a$$$.
The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the elements of array $$$a$$$.
Output format:
For each test case, output a single line containing an integer: the minimum number of operations needed to make all elements in $$$a$$$ equal.
Example Input 0:
7
1
1
3
1 2 3
3
1 2 2
5
5 4 3 2 1
6
1 1 2 2 3 3
8
8 7 6 3 8 7 6 3
6
1 1 4 5 1 4
Example Output 0:
0
2
1
4
4
6
3
Notes:
In the first test case, there is only one element in $$$a$$$, so we can't do any operation.
In the second test case, we can perform the following operations to make all elements in $$$a$$$ equal:
- choose $$$i = 2$$$, delete $$$a_3$$$, then $$$a$$$ would become $$$[1, 2]$$$.
- choose $$$i = 1$$$, delete $$$a_1$$$, then $$$a$$$ would become $$$[2]$$$.
It can be proven that we can't make all elements in $$$a$$$ equal using fewer than $$$2$$$ operations, so the answer is $$$2$$$. | 1 |
Codeforces_test | null | 458 | stdin | [
"6\n3\nabc\n4\nxyyx\n8\nalphabet\n1\nk\n10\naabbccddee\n6\nttbddq\n"
] | [
"cbc\nyyyx\nalphaaet\nk\neabbccddee\ntttddq\n"
] | [
"6\n3\nabc\n4\nxyyx\n8\nalphabet\n1\nk\n10\naabbccddee\n6\nttbddq\n"
] | [
"cbc\nyyyx\nalphaaet\nk\neabbccddee\ntttddq\n"
] | You're given a string $$$s$$$ of length $$$n$$$, consisting of only lowercase English letters.
You must do the following operation exactly once:
- Choose any two indices $$$i$$$ and $$$j$$$ ($$$1 \le i, j \le n$$$). You can choose $$$i = j$$$.
- Set $$$s_i := s_j$$$.
You need to minimize the number of distinct permutations$$$^\dagger$$$ of $$$s$$$. Output any string with the smallest number of distinct permutations after performing exactly one operation.
$$$^\dagger$$$ A permutation of the string is an arrangement of its characters into any order. For example, "bac" is a permutation of "abc" but "bcc" is not.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 500$$$). The description of the test cases follows.
The first line of each test case contains $$$n$$$ ($$$1 \le n \le 10$$$) — the length of string $$$s$$$.
The second line of each test case contains $$$s$$$ of length $$$n$$$. The string contains only lowercase English letters.
Output format:
For each test case, output the required $$$s$$$ after applying exactly one operation. If there are multiple solutions, print any of them.
Example Input 0:
6
3
abc
4
xyyx
8
alphabet
1
k
10
aabbccddee
6
ttbddq
Example Output 0:
cbc
yyyx
alphaaet
k
eabbccddee
tttddq
Notes:
In the first test case, we can obtain the following strings in one operation: "abc", "bbc", "cbc", "aac", "acc", "aba", and "abb".
The string "abc" has $$$6$$$ distinct permutations: "abc", "acb", "bac", "bca", "cab", and "cba".
The string "cbc" has $$$3$$$ distinct permutations: "bcc", "cbc", and "ccb", which is the lowest of all the obtainable strings. In fact, all obtainable strings except "abc" have $$$3$$$ permutations, so any of them would be accepted. | 1 |
Codeforces_test | null | 459 | stdin | [
"2\n5\n\n4\n\n0\n\n1\n\n2\n\n2\n\n0\n"
] | [
"? 1 5\n\n? 2 4\n\n? 4 5\n\n? 3 5\n\n! 01001\n\n? 1 2\n\n! IMPOSSIBLE\n"
] | [
"2\n5\n01001\n2\n11\n"
] | [
"01001\nIMPOSSIBLE\n"
] | This is an interactive problem.
Kachina challenges you to guess her favorite binary string$$$^{\text{∗}}$$$ $$$s$$$ of length $$$n$$$. She defines $$$f(l, r)$$$ as the number of subsequences$$$^{\text{†}}$$$ of $$$\texttt{01}$$$ in $$$s_l s_{l+1} \ldots s_r$$$. Two subsequences are considered different if they are formed by deleting characters from different positions in the original string, even if the resulting subsequences consist of the same characters.
To determine $$$s$$$, you can ask her some questions. In each question, you can choose two indices $$$l$$$ and $$$r$$$ ($$$1 \leq l < r \leq n$$$) and ask her for the value of $$$f(l, r)$$$.
Determine and output $$$s$$$ after asking Kachina no more than $$$n$$$ questions. However, it may be the case that $$$s$$$ is impossible to be determined. In this case, you would need to report $$$\texttt{IMPOSSIBLE}$$$ instead.
Formally, $$$s$$$ is impossible to be determined if after asking $$$n$$$ questions, there are always multiple possible strings for $$$s$$$, regardless of what questions are asked. Note that if you report $$$\texttt{IMPOSSIBLE}$$$ when there exists a sequence of at most $$$n$$$ queries that will uniquely determine the binary string, you will get the Wrong Answer verdict.
Input format:
The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases.
The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 10^4$$$) — the length of $$$s$$$.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^4$$$.
Example Input 0:
2
5
4
0
1
2
2
0
Example Output 0:
? 1 5
? 2 4
? 4 5
? 3 5
! 01001
? 1 2
! IMPOSSIBLE
Notes:
In the first test case:
In the first query, you ask Kachina for the value of $$$f(1, 5)$$$, and she responds with $$$4$$$ in the input stream.
In the second query, you ask Kachina for the value of $$$f(2, 4)$$$. Because there are no subsequences of $$$\texttt{01}$$$ in the string $$$\texttt{100}$$$, she responds with $$$0$$$ in the input stream.
After asking $$$4$$$ questions, you report $$$\texttt{01001}$$$ as $$$s$$$, and it is correct.
In the second test case:
In the first query, you ask Kachina for the value of $$$f(1, 2)$$$, and she responds with $$$0$$$ in the input stream. Notice that this is the only distinct question you can ask.
However, notice that the strings $$$00$$$ and $$$11$$$ both have an answer of $$$0$$$, and it is impossible to differentiate between the two. Therefore, we report IMPOSSIBLE.
Please note that this example only serves to demonstrate the interaction format. It is not guaranteed the queries provided are optimal or uniquely determine the answer. However, it can be shown there exists a sequence of at most $$$5$$$ queries that does uniquely determine sample test case $$$1$$$. | 1 |
Codeforces_test | null | 460 | stdin | [
"11\n1 998244353\n2 998244353\n3 998244353\n4 998244353\n5 998244353\n6 998244353\n7 998244353\n8 998244353\n9 998244353\n10 102275857\n10 999662017\n"
] | [
"0 1 \n1 2 1 \n14 7 4 2 \n183 34 19 16 4 \n2624 209 112 120 48 12 \n42605 1546 793 992 468 216 36 \n785910 13327 6556 9190 4672 2880 864 144 \n16382863 130922 61939 94992 50100 36960 14256 4608 576 \n382823936 1441729 657784 1086596 583344 488700 216000 96480 23040 2880 \n20300780 17572114 7751377 13641280 7376068 6810552 3269700 1785600 576000 144000 14400 \n944100756 17572114 7751377 13641280 7376068 6810552 3269700 1785600 576000 144000 14400\n"
] | [
"11\n1 998244353\n2 998244353\n3 998244353\n4 998244353\n5 998244353\n6 998244353\n7 998244353\n8 998244353\n9 998244353\n10 102275857\n10 999662017\n",
"25\n14 616004887\n16 795653233\n15 156499103\n15 531841069\n16 725817269\n16 959122081\n13 362265263\n18 639275569\n19 303815327\n15 889963901\n11 523951873\n16 457793759\n14 412461589\n18 802848257\n16 959122081\n20 690191011\n11 315480631\n17 581727137\n11 146704721\n17 303815327\n18 407890807\n17 218865187\n15 536563493\n15 828371549\n20 935261221\n",
"1\n2998 480951103\n",
"1\n2999 156499103\n",
"1\n3000 303815327\n",
"4\n623 945591749\n629 122757067\n635 744596999\n722 924445351\n",
"3\n957 281456863\n858 729606287\n921 680737573\n",
"2\n1196 523951873\n1221 202745321\n",
"1\n2047 943469929\n",
"1\n2048 640618673\n",
"1\n2049 685718269\n"
] | [
"0 1 \n1 2 1 \n14 7 4 2 \n183 34 19 16 4 \n2624 209 112 120 48 12 \n42605 1546 793 992 468 216 36 \n785910 13327 6556 9190 4672 2880 864 144 \n16382863 130922 61939 94992 50100 36960 14256 4608 576 \n382823936 1441729 657784 1086596 583344 488700 216000 96480 23040 2880 \n20300780 17572114 7751377 13641280 7376068 6810552 3269700 1785600 576000 144000 14400 \n944100756 17572114 7751377 13641280 7376068 6810552 3269700 1785600 576000 144000 14400 \n",
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210712797 423393486 46349356 260617131 577004711 60669250 340964194 269902037 49340037 93334534 540942946 402879342 235034958 549003465 201496891 662314047 456519932 270692421 463328791 414311012 215282268 155610567 501494121 608446220 324606399 159562002 574924406 437973623 11174731 76321880 416269277 228534000 541480514 393344152 522459095 165082499 227170748 584544866 573149796 673795089 87775449 642179365 178930599 267403590 395226787 223595129 377912666 498479527 541574576 479656924 273970941 640840081 553764860 250910142 124990555 579838838 361438518 514671452 299920604 577034278 136898412 421561512 419688864 186652700 79118244 554838225 238712117 88716541 639510024 338284016 481654617 230050489 165345946 609949862 347745623 215102389 456248239 64729961 660928083 338511518 121640591 28672392 242017730 388910614 514890516 684538306 671054424 303280566 479639496 361041112 536842674 259593975 277144504 402072464 95281194 630193180 167701812 564268637 232550580 466652658 199094185 150524490 133799971 446013246 236348737 452865571 537145885 246277046 129347557 623985898 567277025 623348954 27339914 45267749 542392969 437526188 127564887 512706768 639982954 257821581 470657801 505153326 644918284 57221479 568767997 408168138 297243802 631987317 16556720 620079441 465936181 296285096 96470002 208729280 234617024 296270345 207401586 140147988 490917795 670815381 124507022 641630542 324047335 671769442 502671670 27725252 123802987 155442385 372948635 594479124 574290392 351076804 287722256 22761560 473783694 71605343 129399959 385780688 258342242 590626733 262734176 426394319 578050227 431296931 443162522 262616768 465484806 396940490 431088807 184938369 103413044 233899435 609743485 376674242 557596893 429190005 197135139 386922157 114579267 646948422 578682075 571078423 175124562 451344015 259072743 399655413 375340649 488032138 525417535 401269686 622925654 59862912 508868984 159712960 394123844 584456313 199720484 508864362 601237081 443696288 303785408 507703016 482813218 436402054 262562729 100113077 292728698 85958718 313029011 407621753 343509608 464117233 276844519 298208037 569523371 456159039 424055101 236094173 355542082 139121800 430382975 431578286 481711002 476854024 108321154 684210941 620647275 493708150 68564693 584697922 179427533 29453065 358794217 41218838 587997143 229523694 169059324 541759569 320116773 337291908 344629166 423389452 56282657 678982344 370530198 586040221 280614062 209027310 220321338 46058505 228034212 292581967 583803745 436247530 413036013 468962726 495897277 72182219 338490025 464086293 589906467 622445894 127154171 278601275 203902291 267575354 7833760 59746096 259694574 582834051 443379384 142674774 595649454 494120923 23422153 170576746 100354451 661717825 154146995 74240988 350470168 616142852 666514124 504896255 361769428 331226985 531149456 59651131 542945906 468424490 255825325 503291987 529140081 530347459 550927766 605474402 18995414 373529611 63334642 307424634 245001205 652368128 116024578 296544229 169231899 548591553 353465078 180795802 464000460 5532744 101638110 407133136 515312957 670846205 350142872 329688516 497406584 589229437 179170689 81816005 317834790 31336873 75082985 208933510 136855373 657099843 607265354 131050659 674673706 418029619 10601492 442452956 296538614 261441058 267485400 227748902 563489999 307140828 585834665 459583684 120677886 155920584 401342453 584740564 473657263 613088537 600254756 532534546 570066117 451424702 656719325 450139315 581008953 291688126 81304965 467798757 86875294 162508873 90457293 171583262 410985107 165397707 213829162 582830259 188698227 254737429 162555295 389302808 554207644 587596258 303638742 590079704 524755046 552282348 624425887 16522653 511596911 86404988 322828812 352383805 491887706 349780253 25036848 393178432 191568074 342272152 540253003 82092334 549310202 626769429 310379733 141051428 43544949 380146753 345363478 643316165 108497224 591695270 362554187 290588277 427756545 471836573 253249496 106354132 192420221 19549131 399917687 338674702 493971661 75100206 177246036 28242906 124922361 521128882 108501228 295322623 257082397 287014755 538063754 377910308 381228977 487552947 333584942 295030771 122478760 165072 454060594 356240713 355539291 360435729 666834362 532765583 66049249 476922842 538291582 393481809 529521755 570971393 194460675 120833022 291001538 256529380 143555118 557428635 534745738 321012985 233549549 433601670 327715998 586513895 68989812 593875567 36840649 201126064 132919346 104290478 230825801 179101027 15838626 87657821 211251922 373630893 607434041 170429966 172369926 616746082 433965030 564703354 682824167 287768348 479123699 273609269 363518826 566199075 624509644 553529758 435499123 83200452 451451012 513864396 557473105 544402221 462416935 196350566 315018384 343942319 136614753 19549640 44367986 664509729 631078181 525234352 152295190 182641151 629411619 192854650 188569945 105868469 442700138 318230227 493744451 365551080 388360681 261818520 661879806 571682203 69525684 336921063 47404604 35093356 44841783 478168110 679351865 228298701 288947018 416757248 5276896 431020569 160043622 397715126 16699730 76477007 145616428 628855931 6931020 88341840 503493075 218405341 562850514 440225901 298663121 500419920 221326068 535618455 652177998 278477470 540720866 95998773 620676882 599203619 404164821 676827867 161574011 406829346 354565299 504412171 16022801 672018307 255715937 285291271 97403370 626195010 544687112 529827176 678120065 307403860 161294014 554353690 524396434 586580368 473782506 536895895 156551385 662698909 72742195 373891020 631270494 122307551 294144769 598974457 214248328 561745263 678338466 524331996 306586328 371625952 530940409 618008045 295587638 623451370 96618711 307685174 446615877 157657498 586304350 662838089 616208887 200530148 553566609 600069784 300405261 128670566 39661946 203527056 65685034 557057302 314202852 487842075 576167177 257922005 108243230 179770861 184619877 242630273 250017406 304134414 3417515 554524412 567366638 491079560 333345814 311878735 300311678 65614283 245697492 540958190 584324104 315237061 501548763 171584795 534662034 52638725 119254205 426851316 142635388 399403911 446326877 449492582 206113598 59781463 9739068 \n"
] | This is the hard version of the problem. In the three versions, the constraints on $$$n$$$ and the time limit are different. You can make hacks only if all the versions of the problem are solved.
This is the statement of Problem D1B:
- There are $$$n$$$ cities in a row, numbered $$$1, 2, \ldots, n$$$ left to right. At time $$$1$$$, you conquer exactly one city, called the starting city. At time $$$2, 3, \ldots, n$$$, you can choose a city adjacent to the ones conquered so far and conquer it. You win if, for each $$$i$$$, you conquer city $$$i$$$ at a time no later than $$$a_i$$$. A winning strategy may or may not exist, also depending on the starting city. How many starting cities allow you to win?
For each $$$0 \leq k \leq n$$$, count the number of arrays of positive integers $$$a_1, a_2, \ldots, a_n$$$ such that
- $$$1 \leq a_i \leq n$$$ for each $$$1 \leq i \leq n$$$;
- the answer to Problem D1B is $$$k$$$.
The answer can be very large, so you have to calculate it modulo a given prime $$$p$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 3000$$$). The description of the test cases follows.
The only line of each test case contains two integers $$$n$$$, $$$p$$$ ($$$1 \le n \le 3000$$$, $$$10^8 \leq p \leq 10^9$$$, $$$p$$$ is prime) — the number of cities and the modulo.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3000$$$.
Output format:
For each test case, output $$$n+1$$$ integers: the $$$i$$$-th integer should be the number of arrays that satisfy the conditions for $$$k = i-1$$$.
Example Input 0:
11
1 998244353
2 998244353
3 998244353
4 998244353
5 998244353
6 998244353
7 998244353
8 998244353
9 998244353
10 102275857
10 999662017
Example Output 0:
0 1
1 2 1
14 7 4 2
183 34 19 16 4
2624 209 112 120 48 12
42605 1546 793 992 468 216 36
785910 13327 6556 9190 4672 2880 864 144
16382863 130922 61939 94992 50100 36960 14256 4608 576
382823936 1441729 657784 1086596 583344 488700 216000 96480 23040 2880
20300780 17572114 7751377 13641280 7376068 6810552 3269700 1785600 576000 144000 14400
944100756 17572114 7751377 13641280 7376068 6810552 3269700 1785600 576000 144000 14400
Notes:
In the first test case,
- arrays with $$$1$$$ good starting city: $$$[1]$$$.
In the second test case,
- arrays with $$$0$$$ good starting cities: $$$[1, 1]$$$;
- arrays with $$$1$$$ good starting city: $$$[1, 2]$$$, $$$[2, 1]$$$;
- arrays with $$$2$$$ good starting cities: $$$[2, 2]$$$.
In the third test case,
- arrays with $$$0$$$ good starting cities: $$$[1, 1, 1]$$$, $$$[1, 1, 2]$$$, $$$[1, 1, 3]$$$, $$$[1, 2, 1]$$$, $$$[1, 2, 2]$$$, $$$[1, 3, 1]$$$, $$$[1, 3, 2]$$$, $$$[2, 1, 1]$$$, $$$[2, 1, 2]$$$, $$$[2, 2, 1]$$$, $$$[2, 2, 2]$$$, $$$[2, 3, 1]$$$, $$$[2, 3, 2]$$$, $$$[3, 1, 1]$$$;
- arrays with $$$1$$$ good starting city: $$$[1, 2, 3]$$$, $$$[1, 3, 3]$$$, $$$[2, 1, 3]$$$, $$$[3, 1, 2]$$$, $$$[3, 1, 3]$$$, $$$[3, 2, 1]$$$, $$$[3, 3, 1]$$$;
- arrays with $$$2$$$ good starting cities: $$$[2, 2, 3]$$$, $$$[2, 3, 3]$$$, $$$[3, 2, 2]$$$, $$$[3, 3, 2]$$$;
- arrays with $$$3$$$ good starting cities: $$$[3, 2, 3]$$$, $$$[3, 3, 3]$$$. | 1 |
Codeforces_test | null | 461 | stdin | [
"3\nGARAGE\nGARAGEFORSALE\nABCDE\nAABCD\nTRAINING\nDRAINING\n"
] | [
"14\n10\n16\n"
] | [
"3\nGARAGE\nGARAGEFORSALE\nABCDE\nAABCD\nTRAINING\nDRAINING\n",
"1\nABCD\nDABC\n",
"1\nGUFY\nTFIG\n",
"6\nA\nA\nA\nA\nA\nA\nA\nA\nA\nA\nA\nA\n"
] | [
"14\n10\n16\n",
"8\n",
"8\n",
"2\n2\n2\n2\n2\n2\n"
] | There are two screens which can display sequences of uppercase Latin letters. Initially, both screens display nothing.
In one second, you can do one of the following two actions:
- choose a screen and an uppercase Latin letter, and append that letter to the end of the sequence displayed on that screen;
- choose a screen and copy the sequence from it to the other screen, overwriting the sequence that was displayed on the other screen.
You have to calculate the minimum number of seconds you have to spend so that the first screen displays the sequence $$$s$$$, and the second screen displays the sequence $$$t$$$.
Input format:
The first line contains one integer $$$q$$$ ($$$1 \le q \le 500$$$) — the number of test cases.
Each test case consists of two lines. The first line contains the string $$$s$$$, and the second line contains the string $$$t$$$ ($$$1 \le |s|, |t| \le 100$$$). Both strings consist of uppercase Latin letters.
Output format:
For each test case, print one integer — the minimum possible number of seconds you have to spend so that the first screen displays the sequence $$$s$$$, and the second screen displays the sequence $$$t$$$.
Example Input 0:
3
GARAGE
GARAGEFORSALE
ABCDE
AABCD
TRAINING
DRAINING
Example Output 0:
14
10
16
Notes:
In the first test case, the following sequence of actions is possible:
- spend $$$6$$$ seconds to write the sequence GARAGE on the first screen;
- copy the sequence from the first screen to the second screen;
- spend $$$7$$$ seconds to complete the sequence on the second screen by writing FORSALE.
In the second test case, the following sequence of actions is possible:
- spend $$$1$$$ second to write the sequence A on the second screen;
- copy the sequence from the second screen to the first screen;
- spend $$$4$$$ seconds to complete the sequence on the first screen by writing BCDE;
- spend $$$4$$$ seconds to complete the sequence on the second screen by writing ABCD.
In the third test case, the fastest way to display the sequences is to type both of them character by character without copying, and this requires $$$16$$$ seconds. | 1 |
Codeforces_test | null | 462 | stdin | [
"1\n",
"3\n",
"1000\n"
] | [
"2\n",
"4\n",
"1167\n"
] | [
"1\n",
"3\n",
"1000\n",
"2\n",
"4\n",
"5\n",
"6\n",
"7\n",
"8\n",
"9\n",
"993\n",
"994\n",
"995\n",
"996\n",
"997\n",
"998\n",
"999\n",
"730\n",
"418\n",
"550\n"
] | [
"2\n",
"4\n",
"1167\n",
"3\n",
"5\n",
"6\n",
"7\n",
"9\n",
"10\n",
"11\n",
"1159\n",
"1160\n",
"1161\n",
"1162\n",
"1164\n",
"1165\n",
"1166\n",
"852\n",
"488\n",
"642\n"
] | Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.
All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.
During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.
Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.
Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.
Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?
Input format:
The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.
Output format:
Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.
Example Input 0:
1
Example Output 0:
2
Example Input 1:
3
Example Output 1:
4
Example Input 2:
1000
Example Output 2:
1167
Notes:
In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks. | 1 |
Codeforces_test | null | 463 | stdin | [
"4\n3\n1 2 3\n1 3\n2 3\n4\n3 1 1 3\n1 2\n2 3\n4 2\n4\n2 4 4 2\n1 2\n2 3\n3 4\n13\n1 4 4 7 4 7 1 1 7 11 11 11 11\n1 2\n2 3\n3 4\n4 5\n4 6\n2 7\n7 8\n2 9\n6 10\n5 11\n11 12\n10 13\n"
] | [
"000\n1010\n0001\n1001001000100\n"
] | [
"4\n3\n1 2 3\n1 3\n2 3\n4\n3 1 1 3\n1 2\n2 3\n4 2\n4\n2 4 4 2\n1 2\n2 3\n3 4\n13\n1 4 4 7 4 7 1 1 7 11 11 11 11\n1 2\n2 3\n3 4\n4 5\n4 6\n2 7\n7 8\n2 9\n6 10\n5 11\n11 12\n10 13\n"
] | [
"000\n1010\n0001\n1001001000100\n"
] | Let's define the majority of a sequence of $$$k$$$ elements as the unique value that appears strictly more than $$$\left \lfloor {\frac{k}{2}} \right \rfloor$$$ times. If such a value does not exist, then the sequence does not have a majority. For example, the sequence $$$[1,3,2,3,3]$$$ has a majority $$$3$$$ because it appears $$$3 > \left \lfloor {\frac{5}{2}} \right \rfloor = 2$$$ times, but $$$[1,2,3,4,5]$$$ and $$$[1,3,2,3,4]$$$ do not have a majority.
Skibidus found a tree$$$^{\text{∗}}$$$ of $$$n$$$ vertices and an array $$$a$$$ of length $$$n$$$. Vertex $$$i$$$ has the value $$$a_i$$$ written on it, where $$$a_i$$$ is an integer in the range $$$[1, n]$$$.
For each $$$i$$$ from $$$1$$$ to $$$n$$$, please determine if there exists a non-trivial simple path$$$^{\text{†}}$$$ such that $$$i$$$ is the majority of the sequence of integers written on the vertices that form the path.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 5 \cdot 10^5$$$) — the number of vertices.
The second line of each test case contains $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \le a_i \le n$$$) — the integers written on the vertices.
Each of the next $$$n-1$$$ lines contains two integers $$$u_i$$$ and $$$v_i$$$, denoting the two vertices connected by an edge ($$$1 \le u_i,v_i \le n$$$, $$$u_i \neq v_i$$$).
It is guaranteed that the given edges form a tree.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$5 \cdot 10^5$$$.
Output format:
For each test case, output a binary string $$$s$$$ of length $$$n$$$ on a separate line. $$$s_i$$$ should be computed as follows:
- If there is a non-trivial path containing $$$i$$$ as the majority, $$$s_i$$$ is '1';
- Otherwise, $$$s_i$$$ is '0'.
Example Input 0:
4
3
1 2 3
1 3
2 3
4
3 1 1 3
1 2
2 3
4 2
4
2 4 4 2
1 2
2 3
3 4
13
1 4 4 7 4 7 1 1 7 11 11 11 11
1 2
2 3
3 4
4 5
4 6
2 7
7 8
2 9
6 10
5 11
11 12
10 13
Example Output 0:
000
1010
0001
1001001000100
Notes:
In the first test case, there is no non-trivial path with $$$1$$$, $$$2$$$, or $$$3$$$ as a majority, so the binary string outputted is "000".
In the second test case, $$$1\rightarrow 2\rightarrow 4$$$ is a non-trivial path with $$$3$$$ as a majority. | 1 |
Codeforces_test | null | 464 | stdin | [
"2\n3 2 4\n0 0 0 0\n5 5 7\n0 0 0 0 0 0 0\n"
] | [
"6\n190\n"
] | [
"2\n3 2 4\n0 0 0 0\n5 5 7\n0 0 0 0 0 0 0\n",
"1\n39 1 1\n0\n"
] | [
"6\n190\n",
"1\n"
] | This is the easy version of the problem. The difference between the versions is that in this version, all $$$a_i = 0$$$. You can hack only if you solved all versions of this problem.
There is an $$$n$$$-story building, with floors numbered from $$$1$$$ to $$$n$$$ from bottom to top. There is exactly one person living on each floor.
All the residents of the building have a very important goal today: to launch at least $$$c$$$ paper airplanes collectively. The residents will launch the airplanes in turn. When a person from the $$$i$$$-th floor launches an airplane, all residents on the floors from $$$1$$$ to $$$i$$$ can see it as it descends to the ground.
If, from the perspective of the resident on the $$$i$$$-th floor, at least $$$c$$$ airplanes have already been launched, they will not launch any more airplanes themselves. It is also known that by the end of the day, from the perspective of each resident in the building, at least $$$c$$$ airplanes have been launched, and a total of $$$m$$$ airplanes were thrown.
You carefully monitored this flash mob and recorded which resident from which floor threw each airplane. Unfortunately, the information about who exactly threw some airplanes has been lost. Find the number of ways to fill in the gaps so that the information could be credible. Since the answer can be quite large, output it modulo $$$10^9 + 7$$$.
In this version of the problem, all information has been lost, and the entire array consists of gaps.
It is also possible that you made a mistake in your records, and there is no possible way to restore the gaps. In that case, the answer is considered to be $$$0$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains three integers $$$n, c, m$$$ ($$$1 \le n \le 100$$$, $$$1 \le c \le 100$$$, $$$c \le m \le n \cdot c$$$) — the number of floors in the building, the minimum required number of airplanes, and the number of airplanes actually launched.
The second line of each test case contains $$$m$$$ integers $$$a_1, a_2, \ldots, a_m$$$ ($$$0 \le a_i \le n$$$) — $$$a_i$$$ indicates the resident from which floor launched the $$$i$$$-th airplane; $$$a_i = 0$$$ indicates a gap.
In this version of the problem, it is guaranteed that all $$$a_i = 0$$$.
It is guaranteed that the sum of the values of $$$m$$$ across all test cases does not exceed $$$10^4$$$.
Output format:
For each test case, output the number of ways to fill in the gaps with numbers from $$$1$$$ to $$$n$$$, so that the chronology of the airplane launches could be credible, modulo $$$10^9 + 7$$$.
Example Input 0:
2
3 2 4
0 0 0 0
5 5 7
0 0 0 0 0 0 0
Example Output 0:
6
190
Notes:
In the first test example, all six possible ways to fill in the gaps are as follows:
1. $$$[1, 1, 3, 3]$$$
2. $$$[1, 2, 3, 3]$$$
3. $$$[1, 3, 2, 3]$$$
4. $$$[2, 1, 3, 3]$$$
5. $$$[2, 2, 3, 3]$$$
6. $$$[3, 1, 2, 3]$$$
Note that the array $$$[2, 3, 1, 3]$$$ is not a valid way to fill in the gaps, as the third airplane could not have been launched by the person on the $$$1$$$st floor, since from their perspective, $$$c = 2$$$ airplanes had already been launched.
Also, the array $$$[1, 1, 2, 3]$$$ is not a valid way to fill in the gaps, as from the perspective of the person on the $$$3$$$rd floor, only $$$1$$$ airplane has been launched, while $$$c = 2$$$. | 1 |
Codeforces_test | null | 465 | stdin | [
"5\n5\n1 2 3 4 5\n4\n4 3 2 1\n4\n4 5 2 3\n8\n4 5 4 5 4 5 4 5\n9\n9 9 8 2 4 4 3 5 3\n"
] | [
"YES\nNO\nYES\nYES\nNO\n"
] | [
"5\n5\n1 2 3 4 5\n4\n4 3 2 1\n4\n4 5 2 3\n8\n4 5 4 5 4 5 4 5\n9\n9 9 8 2 4 4 3 5 3\n",
"1\n3\n1 1 2\n"
] | [
"YES\nNO\nYES\nYES\nNO\n",
"YES\n"
] | You are given a sequence $$$a$$$ consisting of $$$n$$$ positive integers.
You can perform the following operation any number of times.
- Select an index $$$i$$$ ($$$1 \le i < n$$$), and subtract $$$\min(a_i,a_{i+1})$$$ from both $$$a_i$$$ and $$$a_{i+1}$$$.
Determine if it is possible to make the sequence non-decreasing by using the operation any number of times.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 2 \cdot 10^5$$$).
The second line of each test case contains $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \le a_i \le 10^9$$$).
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
If it is possible to make the sequence non-decreasing, print "YES" on a new line. Otherwise, print "NO" on a new line.
You can output the answer in any case. For example, the strings "yEs", "yes", and "Yes" will also be recognized as positive responses.
Example Input 0:
5
5
1 2 3 4 5
4
4 3 2 1
4
4 5 2 3
8
4 5 4 5 4 5 4 5
9
9 9 8 2 4 4 3 5 3
Example Output 0:
YES
NO
YES
YES
NO
Notes:
In the first test case, the array is already sorted.
In the second test case, we can show that it is impossible.
In the third test case, after performing an operation on $$$i=1$$$, the array becomes $$$[0,1,2,3]$$$, which is now in nondecreasing order. | 1 |
Codeforces_test | null | 466 | stdin | [
"2\n\n18\n\n25\n\n\n9999\n"
] | [
"? 3 5\n\n? 4 4\n\n! 4\n? 99 100\n\n! 100\n"
] | [
"2\n4\n100\n"
] | [
"4\n100\n"
] | This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.
This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.
We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:
- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.
The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.
You need to find the value of $$$x$$$. To do that, you can make queries of the following form:
- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.
Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.
Input format:
Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.
Example Input 0:
2
18
25
9999
Example Output 0:
? 3 5
? 4 4
! 4
? 99 100
! 100
Notes:
In the first test, the interaction proceeds as follows.
SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.
Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction. | 1 |
Codeforces_test | null | 467 | stdin | [
"9\n2 1\n4 5\n9\n2 1\n3 6\n9\n4 2\n1 2 2 2\n3 4\n4 2\n1 1 3 3\n3 5\n4 2\n1 2 3 4\n3 5\n5 5\n1 2 3 4 5\n5 4 3 2 1\n4 2\n1 1 1 1\n1 1\n4 4\n1 1 1 1\n1 1 1 2\n1 1\n1\n1000000000\n"
] | [
"Yes\nNo\nYes\nYes\nNo\nYes\nNo\nNo\nNo\n"
] | [
"9\n2 1\n4 5\n9\n2 1\n3 6\n9\n4 2\n1 2 2 2\n3 4\n4 2\n1 1 3 3\n3 5\n4 2\n1 2 3 4\n3 5\n5 5\n1 2 3 4 5\n5 4 3 2 1\n4 2\n1 1 1 1\n1 1\n4 4\n1 1 1 1\n1 1 1 2\n1 1\n1\n1000000000\n",
"4\n1 1\n1000000000\n1000000000\n2 2\n500000000 500000000\n1000000000 1000000000\n3 1\n1 500000000 500000000\n1000000000\n8 1\n536870913 536870913 536870913 536870913 536870913 536870913 536870913 536870913\n8\n"
] | [
"Yes\nNo\nYes\nYes\nNo\nYes\nNo\nNo\nNo\n",
"Yes\nNo\nNo\nNo\n"
] | Kevin wrote an integer sequence $$$a$$$ of length $$$n$$$ on the blackboard.
Kevin can perform the following operation any number of times:
- Select two integers $$$x, y$$$ on the blackboard such that $$$|x - y| \leq 1$$$, erase them, and then write down an integer $$$x + y$$$ instead.
Kevin wants to know if it is possible to transform these integers into an integer sequence $$$b$$$ of length $$$m$$$ through some sequence of operations.
Two sequences $$$a$$$ and $$$b$$$ are considered the same if and only if their multisets are identical. In other words, for any number $$$x$$$, the number of times it appears in $$$a$$$ must be equal to the number of times it appears in $$$b$$$.
Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1\leq m \leq n \leq 2\cdot 10^5$$$) — the length of $$$a$$$ and the length of $$$b$$$.
The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1\leq a_i \leq 10^9$$$).
The third line contains $$$m$$$ integers $$$b_1, b_2, \ldots, b_m$$$ ($$$1\leq b_i \leq 10^9$$$).
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2\cdot 10^5$$$.
Output format:
For each test case, output "Yes" if it is possible to transform $$$a$$$ into $$$b$$$, and "No" otherwise.
You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.
Example Input 0:
9
2 1
4 5
9
2 1
3 6
9
4 2
1 2 2 2
3 4
4 2
1 1 3 3
3 5
4 2
1 2 3 4
3 5
5 5
1 2 3 4 5
5 4 3 2 1
4 2
1 1 1 1
1 1
4 4
1 1 1 1
1 1 1 2
1 1
1
1000000000
Example Output 0:
Yes
No
Yes
Yes
No
Yes
No
No
No
Notes:
In the first test case, you can erase $$$4, 5$$$, and write down $$$9$$$.
In the second test case, you can't erase $$$3, 6$$$.
In the third test case, one possible way could be:
- Erase $$$2, 2$$$, and write down $$$4$$$. The remaining numbers are $$$1, 2, 4$$$ now.
- Erase $$$1, 2$$$, and write down $$$3$$$. The remaining numbers are $$$3, 4$$$ now.
In the fourth test case, one possible way could be:
- Erase $$$1, 1$$$, and write down $$$2$$$. The remaining numbers are $$$2, 3, 3$$$ now.
- Erase $$$2, 3$$$, and write down $$$5$$$. The remaining numbers are $$$3, 5$$$ now. | 1 |
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