knightnemo/LMCompressionDataset · Datasets at Hugging Face

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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 $0$.
Kosuke is too lazy. He will not give you any legend, just the task:

Fibonacci numbers are defined as follows:

* $f(1)=f(2)=1$. * $f(n)=f(n-1)+f(n-2)$ $(3\le n)$

We denote $G(n,k)$ as an index of the $n$-th Fibonacci number that is divisible by $k$. For given $n$ and $k$, compute $G(n,k)$.

As this number can be too big, output it by modulo $10^9+7$.

For example: $G(3,2)=9$ because the $3$-rd Fibonacci number that is divisible by $2$ is $34$. $[1,1,\textbf{2},3,5,\textbf{8},13,21,\textbf{34}]$.

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

The first and only line contains two integers $n$ and $k$ ($1 \le n \le 10^{18}$, $1 \le k \le 10^5$).

It is guaranteed that the sum of $k$ across all test cases does not exceed $10^6$.

For each test case, output the only number: the value $G(n,k)$ taken by modulo $10^9+7$.

Red was ejected. They were not the imposter.

There are $n$ rows of $m$ people. Let the position in the $r$-th row and the $c$-th column be denoted by $(r, c)$. Number each person starting from $1$ in row-major order, i.e., the person numbered $(r-1)\cdot m+c$ is initially at $(r,c)$.

The person at $(r, c)$ decides to leave. To fill the gap, let the person who left be numbered $i$. Each person numbered $j>i$ will move to the position where the person numbered $j-1$ is initially at. The following diagram illustrates the case where $n=2$, $m=3$, $r=1$, and $c=2$.

![](CDN_BASE_URL/9b0b8e601446e3410296d7c9b1ff8763)

Calculate the sum of the Manhattan distances of each person's movement. If a person was initially at $(r_0, c_0)$ and then moved to $(r_1, c_1)$, the Manhattan distance is $\|r_0-r_1\|+\|c_0-c_1\|$.

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

The only line of each testcase contains $4$ integers $n$, $m$, $r$, and $c$ ($1\le r\le n\le 10^6$, $1 \le c \le m \le 10^6$), where $n$ is the number of rows, $m$ is the number of columns, and $(r,c)$ is the position where the person who left is initially at.

For each test case, output a single integer denoting the sum of the Manhattan distances.

For the first test case, the person numbered $2$ leaves, and the distances of the movements of the person numbered $3$, $4$, $5$, and $6$ are $1$, $3$, $1$, and $1$, respectively. So the answer is $1+3+1+1=6$.

For the second test case, the person numbered $3$ leaves, and the person numbered $4$ moves. The answer is $1$.
There are 3 heroes and 3 villains, so 6 people in total.

Given a positive integer $n$. Find the smallest integer whose decimal representation has length $n$ and consists only of $3$s and $6$s such that it is divisible by both $33$ and $66$. If no such integer exists, print $-1$.

The first line contains a single integer $t$ ($1\le t\le 500$) — the number of test cases.

The only line of each test case contains a single integer $n$ ($1\le n\le 500$) — the length of the decimal representation.

For each test case, output the smallest required integer if such an integer exists and $-1$ otherwise.

For $n=1$, no such integer exists as neither $3$ nor $6$ is divisible by $33$.

For $n=2$, $66$ consists only of $6$s and it is divisible by both $33$ and $66$.

For $n=3$, no such integer exists. Only $363$ is divisible by $33$, but it is not divisible by $66$.

For $n=4$, $3366$ and $6666$ are divisible by both $33$ and $66$, and $3366$ is the smallest.
Man, this Genshin boss is so hard. Good thing they have a top-up of $6$ coins for only $ \$4.99$. I should be careful and spend no more than I need to, lest my mom catches me...

You are fighting a monster with $z$ health using a weapon with $d$ damage. Initially, $d=0$. You can perform the following operations.

* Increase $d$ — the damage of your weapon by $1$, costing $x$ coins. * Attack the monster, dealing $d$ damage and costing $y$ coins.

You cannot perform the first operation for more than $k$ times in a row.

Find the minimum number of coins needed to defeat the monster by dealing at least $z$ damage.

The first line contains a single integer $t$ ($1\le t\le 100$) — the number of test cases.

The only line of each test case contains 4 integers $x$, $y$, $z$, and $k$ ($1\leq x, y, z, k\leq 10^8$) — the first operation's cost, the second operation's cost, the monster's health, and the limitation on the first operation.

For each test case, output the minimum number of coins needed to defeat the monster.

In the first test case, $x = 2$, $y = 3$, $z = 5$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $12$ coins:

* Increase damage by $1$, costing $2$ coins. * Increase damage by $1$, costing $2$ coins. * Increase damage by $1$, costing $2$ coins. * Attack the monster, dealing $3$ damage, costing $3$ coins. * Attack the monster, dealing $3$ damage, costing $3$ coins.

You deal a total of $3 + 3 = 6$ damage, defeating the monster who has $5$ health. The total number of coins you use is $2 + 2 + 2 + 3 + 3 = 12$ coins.

In the second test case, $x = 10$, $y = 20$, $z = 40$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $190$ coins:

* Increase damage by $5$, costing $5\cdot x$ = $50$ coins. * Attack the monster once, dealing $5$ damage, costing $20$ coins. * Increase damage by $2$, costing $2\cdot x$ = $20$ coins. * Attack the monster $5$ times, dealing $5\cdot 7 = 35$ damage, costing $5\cdot y$ = $100$ coins.

You deal
Monocarp is opening his own IT company. He wants to hire $n$ programmers and $m$ testers.

There are $n+m+1$ candidates, numbered from $1$ to $n+m+1$ in chronological order of their arriving time. The $i$-th candidate has programming skill $a_i$ and testing skill $b_i$ (a person's programming skill is different from their testing skill). The skill of the team is the sum of the programming skills of all candidates hired as programmers, and the sum of the testing skills of all candidates hired as testers.

When a candidate arrives to interview, Monocarp tries to assign them to the most suitable position for them (if their programming skill is higher, then he hires them as a programmer, otherwise as a tester). If all slots for that position are filled, Monocarp assigns them to the other position.

Your task is, for each candidate, calculate the skill of the team if everyone except them comes to interview. Note that it means that exactly $n+m$ candidates will arrive, so all $n+m$ positions in the company will be filled.

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

Each test case consists of three lines:

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 $0$.

Kosuke is too lazy. He will not give you any legend, just the task:

Fibonacci numbers are defined as follows:

* $f(1)=f(2)=1$. * $f(n)=f(n-1)+f(n-2)$ $(3\le n)$

We denote $G(n,k)$ as an index of the $n$-th Fibonacci number that is divisible by $k$. For given $n$ and $k$, compute $G(n,k)$.

As this number can be too big, output it by modulo $10^9+7$.

For example: $G(3,2)=9$ because the $3$-rd Fibonacci number that is divisible by $2$ is $34$. $[1,1,\textbf{2},3,5,\textbf{8},13,21,\textbf{34}]$.

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

The first and only line contains two integers $n$ and $k$ ($1 \le n \le 10^{18}$, $1 \le k \le 10^5$).

It is guaranteed that the sum of $k$ across all test cases does not exceed $10^6$.

For each test case, output the only number: the value $G(n,k)$ taken by modulo $10^9+7$.

Red was ejected. They were not the imposter.

There are $n$ rows of $m$ people. Let the position in the $r$-th row and the $c$-th column be denoted by $(r, c)$. Number each person starting from $1$ in row-major order, i.e., the person numbered $(r-1)\cdot m+c$ is initially at $(r,c)$.

The person at $(r, c)$ decides to leave. To fill the gap, let the person who left be numbered $i$. Each person numbered $j>i$ will move to the position where the person numbered $j-1$ is initially at. The following diagram illustrates the case where $n=2$, $m=3$, $r=1$, and $c=2$.

![](CDN_BASE_URL/9b0b8e601446e3410296d7c9b1ff8763)

Calculate the sum of the Manhattan distances of each person's movement. If a person was initially at $(r_0, c_0)$ and then moved to $(r_1, c_1)$, the Manhattan distance is $|r_0-r_1|+|c_0-c_1|$.

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

The only line of each testcase contains $4$ integers $n$, $m$, $r$, and $c$ ($1\le r\le n\le 10^6$, $1 \le c \le m \le 10^6$), where $n$ is the number of rows, $m$ is the number of columns, and $(r,c)$ is the position where the person who left is initially at.

For each test case, output a single integer denoting the sum of the Manhattan distances.

For the first test case, the person numbered $2$ leaves, and the distances of the movements of the person numbered $3$, $4$, $5$, and $6$ are $1$, $3$, $1$, and $1$, respectively. So the answer is $1+3+1+1=6$.

For the second test case, the person numbered $3$ leaves, and the person numbered $4$ moves. The answer is $1$.

There are 3 heroes and 3 villains, so 6 people in total.

Given a positive integer $n$. Find the smallest integer whose decimal representation has length $n$ and consists only of $3$s and $6$s such that it is divisible by both $33$ and $66$. If no such integer exists, print $-1$.

The first line contains a single integer $t$ ($1\le t\le 500$) — the number of test cases.

The only line of each test case contains a single integer $n$ ($1\le n\le 500$) — the length of the decimal representation.

For each test case, output the smallest required integer if such an integer exists and $-1$ otherwise.

For $n=1$, no such integer exists as neither $3$ nor $6$ is divisible by $33$.

For $n=2$, $66$ consists only of $6$s and it is divisible by both $33$ and $66$.

For $n=3$, no such integer exists. Only $363$ is divisible by $33$, but it is not divisible by $66$.

For $n=4$, $3366$ and $6666$ are divisible by both $33$ and $66$, and $3366$ is the smallest.

Man, this Genshin boss is so hard. Good thing they have a top-up of $6$ coins for only $ \$4.99$. I should be careful and spend no more than I need to, lest my mom catches me...

You are fighting a monster with $z$ health using a weapon with $d$ damage. Initially, $d=0$. You can perform the following operations.

* Increase $d$ — the damage of your weapon by $1$, costing $x$ coins. * Attack the monster, dealing $d$ damage and costing $y$ coins.

You cannot perform the first operation for more than $k$ times in a row.

Find the minimum number of coins needed to defeat the monster by dealing at least $z$ damage.

The first line contains a single integer $t$ ($1\le t\le 100$) — the number of test cases.

The only line of each test case contains 4 integers $x$, $y$, $z$, and $k$ ($1\leq x, y, z, k\leq 10^8$) — the first operation's cost, the second operation's cost, the monster's health, and the limitation on the first operation.

For each test case, output the minimum number of coins needed to defeat the monster.

In the first test case, $x = 2$, $y = 3$, $z = 5$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $12$ coins:

* Increase damage by $1$, costing $2$ coins. * Increase damage by $1$, costing $2$ coins. * Increase damage by $1$, costing $2$ coins. * Attack the monster, dealing $3$ damage, costing $3$ coins. * Attack the monster, dealing $3$ damage, costing $3$ coins.

You deal a total of $3 + 3 = 6$ damage, defeating the monster who has $5$ health. The total number of coins you use is $2 + 2 + 2 + 3 + 3 = 12$ coins.

In the second test case, $x = 10$, $y = 20$, $z = 40$, and $k = 5$. Here's a strategy that achieves the lowest possible cost of $190$ coins:

* Increase damage by $5$, costing $5\cdot x$ = $50$ coins. * Attack the monster once, dealing $5$ damage, costing $20$ coins. * Increase damage by $2$, costing $2\cdot x$ = $20$ coins. * Attack the monster $5$ times, dealing $5\cdot 7 = 35$ damage, costing $5\cdot y$ = $100$ coins.

You deal

Monocarp is opening his own IT company. He wants to hire $n$ programmers and $m$ testers.

There are $n+m+1$ candidates, numbered from $1$ to $n+m+1$ in chronological order of their arriving time. The $i$-th candidate has programming skill $a_i$ and testing skill $b_i$ (a person's programming skill is different from their testing skill). The skill of the team is the sum of the programming skills of all candidates hired as programmers, and the sum of the testing skills of all candidates hired as testers.

When a candidate arrives to interview, Monocarp tries to assign them to the most suitable position for them (if their programming skill is higher, then he hires them as a programmer, otherwise as a tester). If all slots for that position are filled, Monocarp assigns them to the other position.

Your task is, for each candidate, calculate the skill of the team if everyone except them comes to interview. Note that it means that exactly $n+m$ candidates will arrive, so all $n+m$ positions in the company will be filled.

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

Each test case consists of three lines: