knightnemo/LMCompressionDataset · Datasets at Hugging Face

text stringlengths 0 801
* the first line contains two integers $n$ and $m$ ($0 \le n, m \le 2 \cdot 10^5$; $2 \le n + m + 1 \le 2 \cdot 10^5$) — the number of programmers and the number of testers Monocarp wants to hire, respectively; * the second line contains $n + m + 1$ integers $a_1, a_2, \dots, a_{n+m+1}$ ($1 \le a_i \le 10^9$), where $a_i$ is the programming skill of the $i$-th candidate; * the third line contains $n + m + 1$ integers $b_1, b_2, \dots, b_{n+m+1}$ ($1 \le b_i \le 10^9$; $b_i \ne a_i$), where $b_i$ is the testing skill of the $i$-th candidate.

Additional constraint on the input: the sum of $(n + m + 1)$ over all test cases doesn't exceed $2 \cdot 10^5$.

For each test case, print $n + m + 1$ integers, where the $i$-th integer should be equal to the skill of the team if everyone except the $i$-th candidate comes to interview.

Let's consider the third test
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](http://tiny.cc/xor_wiki_eng).

Since your magical abilities as a librarian are severely limited, you can make no more than $150$ queries.

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.



In the first test case, the books in the library afte
You are given a positive integer $x$. Find any array of integers $a_0, a_1, \ldots, a_{n-1}$ for which the following holds:

* $1 \le n \le 32$, * $a_i$ is $1$, $0$, or $-1$ for all $0 \le i \le n - 1$, * $x = \displaystyle{\sum_{i=0}^{n - 1}{a_i \cdot 2^i}}$, * There does not exist an index $0 \le i \le n - 2$ such that both $a_{i} \neq 0$ and $a_{i + 1} \neq 0$.

It can be proven that under the constraints of the problem, a valid array always exists.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains a single positive integer $x$ ($1 \le x < 2^{30}$).

For each test case, output two lines.

On the first line, output an integer $n$ ($1 \le n \le 32$) — the length of the array $a_0, a_1, \ldots, a_{n-1}$.

On the second line, output the array $a_0, a_1, \ldots, a_{n-1}$.

If there are multiple valid arrays, you can output any of them.

In the first test case, one valid array is $[1]$, since $(1) \cdot 2^0 = 1$.

In the second test case, one possible valid array is $[0,-1,0,0,1]$, since $(0) \cdot 2^0 + (-1) \cdot 2^1 + (0) \cdot 2^2 + (0) \cdot 2^3 + (1) \cdot 2^4 = -2 + 16 = 14$.
Timur is in a car traveling on the number line from point $0$ to point $n$. The car starts moving from point $0$ at minute $0$.

There are $k+1$ signs on the line at points $0, a_1, a_2, \dots, a_k$, and Timur knows that the car will arrive there at minutes $0, b_1, b_2, \dots, b_k$, respectively. The sequences $a$ and $b$ are strictly increasing with $a_k = n$.

![](CDN_BASE_URL/bacc58f5d5013f5922e96a1311487b97)

Between any two adjacent signs, the car travels with a constant speed. Timur has $q$ queries: each query will be an integer $d$, and Timur wants you to output how many minutes it takes the car to reach point $d$, rounded down to the nearest integer.

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 three integers $n$, $k$, and $q$, ($k \leq n \leq 10^9$; $1 \leq k, q \leq 10^5$) — the final destination, the number of points Timur knows the time for, and the number of queries respectively.

The second line of each test case contains $k$ integers $a_i$ ($1 \leq a_i \leq n$; $a_i < a_{i+1}$ for every $1 \leq i \leq k-1$; $a_k = n$).

The third line of each test case contains $k$ integers $b_i$ ($1 \leq b_i \leq 10^9$; $b_i < b_{i+1}$ for every $1 \leq i \leq k-1$).

Each of the following $q$ lines contains a single integer $d$ ($0 \leq d \leq n$) — the distance that Timur asks the minutes passed for.

The sum of $k$ over all test cases doesn't exceed $10^5$, and the sum of $q$ over all test cases doesn't exceed $10^5$.

For each query, output a single integer — the number of minutes passed until the car reaches the point $d$, rounded down.

For the first test case, the car goes from point $0$ to point $10$ in $10$ minutes, so the speed is $1$ unit per minute and:

* At point $0$, the time will be $0$ minutes. * At point $6$, the time will be $6$ minutes. * At point $7$, the time will be $7$ minutes.

For the second test case, between points $0$ and $4$, the car travels at a speed of $1$ unit per minut
Bob decided to open a bakery. On the opening day, he baked $n$ buns that he can sell. The usual price of a bun is $a$ coins, but to attract customers, Bob organized the following promotion:

* Bob chooses some integer $k$ ($0 \le k \le \min(n, b)$). * Bob sells the first $k$ buns at a modified price. In this case, the price of the $i$-th ($1 \le i \le k$) sold bun is $(b - i + 1)$ coins. * The remaining $(n - k)$ buns are sold at $a$ coins each.

Note that $k$ can be equal to $0$. In this case, Bob will sell all the buns at $a$ coins each.

Help Bob determine the maximum profit he can obtain by selling all $n$ buns.

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 only line of each test case contains three integers $n$, $a$, and $b$ ($1 \le n, a, b \le 10^9$) — the number of buns, the usual price of a bun, and the price of the first bun to be sold at a modified price.

For each test case, output a single integer — the maximum profit that Bob can obtain.

In the first test case, it is optimal for Bob to choose $k = 1$. Then he will sell one bun for $5$ coins, and three buns at the usual price for $4$ coins each. Then the profit will be $5 + 4 + 4 + 4 = 17$ coins.

In the second test case, it is optimal for Bob to choose $k = 5$. Then he will sell all the buns at the modified price and obtain a profit of $9 + 8 + 7 + 6 + 5 = 35$ coins.

In the third test case, it is optimal for Bob to choose $k = 0$. Then he will sell all the buns at the usual price and obtain a profit of $10 \cdot 10 = 100$ coins.
Dmitry has $n$ cubes, numbered from left to right from $1$ to $n$. The cube with index $f$ is his favorite.

Dmitry threw all the cubes on the table, and the $i$-th cube showed the value $a_i$ ($1 \le a_i \le 100$). After that, he arranged the cubes in non-increasing order of their values, from largest to smallest. If two cubes show the same value, they can go in any order.

After sorting, Dmitry removed the first $k$ cubes. Then he became interested in whether he removed his favorite cube (note that its position could have changed after sorting).

* the first line contains two integers $n$ and $m$ ($0 \le n, m \le 2 \cdot 10^5$; $2 \le n + m + 1 \le 2 \cdot 10^5$) — the number of programmers and the number of testers Monocarp wants to hire, respectively; * the second line contains $n + m + 1$ integers $a_1, a_2, \dots, a_{n+m+1}$ ($1 \le a_i \le 10^9$), where $a_i$ is the programming skill of the $i$-th candidate; * the third line contains $n + m + 1$ integers $b_1, b_2, \dots, b_{n+m+1}$ ($1 \le b_i \le 10^9$; $b_i \ne a_i$), where $b_i$ is the testing skill of the $i$-th candidate.

Additional constraint on the input: the sum of $(n + m + 1)$ over all test cases doesn't exceed $2 \cdot 10^5$.

For each test case, print $n + m + 1$ integers, where the $i$-th integer should be equal to the skill of the team if everyone except the $i$-th candidate comes to interview.

Let's consider the third test

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](http://tiny.cc/xor_wiki_eng).

Since your magical abilities as a librarian are severely limited, you can make no more than $150$ queries.

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.

In the first test case, the books in the library afte

You are given a positive integer $x$. Find any array of integers $a_0, a_1, \ldots, a_{n-1}$ for which the following holds:

* $1 \le n \le 32$, * $a_i$ is $1$, $0$, or $-1$ for all $0 \le i \le n - 1$, * $x = \displaystyle{\sum_{i=0}^{n - 1}{a_i \cdot 2^i}}$, * There does not exist an index $0 \le i \le n - 2$ such that both $a_{i} \neq 0$ and $a_{i + 1} \neq 0$.

It can be proven that under the constraints of the problem, a valid array always exists.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains a single positive integer $x$ ($1 \le x < 2^{30}$).

For each test case, output two lines.

On the first line, output an integer $n$ ($1 \le n \le 32$) — the length of the array $a_0, a_1, \ldots, a_{n-1}$.

On the second line, output the array $a_0, a_1, \ldots, a_{n-1}$.

If there are multiple valid arrays, you can output any of them.

In the first test case, one valid array is $[1]$, since $(1) \cdot 2^0 = 1$.

In the second test case, one possible valid array is $[0,-1,0,0,1]$, since $(0) \cdot 2^0 + (-1) \cdot 2^1 + (0) \cdot 2^2 + (0) \cdot 2^3 + (1) \cdot 2^4 = -2 + 16 = 14$.

Timur is in a car traveling on the number line from point $0$ to point $n$. The car starts moving from point $0$ at minute $0$.

There are $k+1$ signs on the line at points $0, a_1, a_2, \dots, a_k$, and Timur knows that the car will arrive there at minutes $0, b_1, b_2, \dots, b_k$, respectively. The sequences $a$ and $b$ are strictly increasing with $a_k = n$.

![](CDN_BASE_URL/bacc58f5d5013f5922e96a1311487b97)

Between any two adjacent signs, the car travels with a constant speed. Timur has $q$ queries: each query will be an integer $d$, and Timur wants you to output how many minutes it takes the car to reach point $d$, rounded down to the nearest integer.

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 three integers $n$, $k$, and $q$, ($k \leq n \leq 10^9$; $1 \leq k, q \leq 10^5$) — the final destination, the number of points Timur knows the time for, and the number of queries respectively.

The second line of each test case contains $k$ integers $a_i$ ($1 \leq a_i \leq n$; $a_i < a_{i+1}$ for every $1 \leq i \leq k-1$; $a_k = n$).

The third line of each test case contains $k$ integers $b_i$ ($1 \leq b_i \leq 10^9$; $b_i < b_{i+1}$ for every $1 \leq i \leq k-1$).

Each of the following $q$ lines contains a single integer $d$ ($0 \leq d \leq n$) — the distance that Timur asks the minutes passed for.

The sum of $k$ over all test cases doesn't exceed $10^5$, and the sum of $q$ over all test cases doesn't exceed $10^5$.

For each query, output a single integer — the number of minutes passed until the car reaches the point $d$, rounded down.

For the first test case, the car goes from point $0$ to point $10$ in $10$ minutes, so the speed is $1$ unit per minute and:

* At point $0$, the time will be $0$ minutes. * At point $6$, the time will be $6$ minutes. * At point $7$, the time will be $7$ minutes.

For the second test case, between points $0$ and $4$, the car travels at a speed of $1$ unit per minut

Bob decided to open a bakery. On the opening day, he baked $n$ buns that he can sell. The usual price of a bun is $a$ coins, but to attract customers, Bob organized the following promotion:

* Bob chooses some integer $k$ ($0 \le k \le \min(n, b)$). * Bob sells the first $k$ buns at a modified price. In this case, the price of the $i$-th ($1 \le i \le k$) sold bun is $(b - i + 1)$ coins. * The remaining $(n - k)$ buns are sold at $a$ coins each.

Note that $k$ can be equal to $0$. In this case, Bob will sell all the buns at $a$ coins each.

Help Bob determine the maximum profit he can obtain by selling all $n$ buns.

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 only line of each test case contains three integers $n$, $a$, and $b$ ($1 \le n, a, b \le 10^9$) — the number of buns, the usual price of a bun, and the price of the first bun to be sold at a modified price.

For each test case, output a single integer — the maximum profit that Bob can obtain.

In the first test case, it is optimal for Bob to choose $k = 1$. Then he will sell one bun for $5$ coins, and three buns at the usual price for $4$ coins each. Then the profit will be $5 + 4 + 4 + 4 = 17$ coins.

In the second test case, it is optimal for Bob to choose $k = 5$. Then he will sell all the buns at the modified price and obtain a profit of $9 + 8 + 7 + 6 + 5 = 35$ coins.

In the third test case, it is optimal for Bob to choose $k = 0$. Then he will sell all the buns at the usual price and obtain a profit of $10 \cdot 10 = 100$ coins.

Dmitry has $n$ cubes, numbered from left to right from $1$ to $n$. The cube with index $f$ is his favorite.

Dmitry threw all the cubes on the table, and the $i$-th cube showed the value $a_i$ ($1 \le a_i \le 100$). After that, he arranged the cubes in non-increasing order of their values, from largest to smallest. If two cubes show the same value, they can go in any order.

After sorting, Dmitry removed the first $k$ cubes. Then he became interested in whether he removed his favorite cube (note that its position could have changed after sorting).