meetforever/GRPO_CL huggingface dataset:
user: After the hundred years of war started by the Fire Nation, its time for the Water Tribes to bring it to an end. Avatar asked Sokka to go to the Southern WarZone from The Northern WarZone and gave him some money in a bag for his journey. Sokka has the World Map to guide him during his journey . The World Map is described as a $N$x$N$ grid where the Northern WarZone is denoted by $(0,0)$ and the Southern WarZone is denoted by $(N-1,N-1)$. Each location on the world map is denoted in a similar way by two integers r and c. For each location having:  
- $r = c$ denotes neutral land ( $(0,0)$ and $(N-1,N-1)$ also come under this category)       
- $r < c$ denotes the land under the rule of Water Tribes     
- $r > c$ denotes the land under the rule of Fire Kingdom   
Being wise Sokka travels only from one location to any other valid location (by valid location we mean a location which exists on the world map grid i.e. for that location $0 \leq r < N$ and $0 \leq c < N$ ) just to the right $(r,c+1)$ or below $(r+1,c)$ the current location randomly. Due to the times of war , Sokka has to pay one coin  each time he transitions from one nation to another. Here a transition is counted when Sokka is in Water Tribe land and moves to Fire Nation Land crossing the neutral land or vice versa .The first move is obviously never counted as a transition. Moreover , the coin is to be payed exactly once for one such transition (eg. if he makes this transition k times he has to pay k coins).  
The initial number of coins Sokka has is $2*N$. The probability that the coins he has when he reaches his destination is lesser than the number of coins he started his journey with can be expressed as a fraction $P/Q$, where P and Q are integers $(P \geq 0, Q > 0)$  and Q is co-prime with $(10^9)+7$. You should compute $P/Q$ modulo $(10^9)+7$ for $T$ values of $N$.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase contains of a single line of input, one integer $N$, the size of the world map grid. 

-----Output:-----
For each testcase, output in a single line , the Probability modulo (10^9)+7.

-----Constraints-----
- $1 \leq T \leq 100000$
- $2 \leq N \leq 10^9$

-----Sample Input:-----
2    

5

2    

-----Sample Output:-----
200000002     

0  

-----EXPLANATION:-----
For N=2, there is no possible path in which Sokka has to spend money, so the final number of coins in the bag is not lesser than the initial amount.
user: Bandwidth of a matrix A is defined as the smallest non-negative integer K such that A(i, j) = 0 for |i - j| > K.
For example, a matrix with all zeros will have its bandwith equal to zero. Similarly bandwith of diagonal matrix will also be zero.

For example, for the below given matrix, the bandwith of this matrix is 2.

1 0 0
0 1 1
1 1 0 

Bandwidth of the below matrix is 1. 

Bandwidth of the below matrix is 2. 

Bandwidth of the below matrix is also 2. 

You will be a given a binary matrix A of dimensions N × N. You are allowed to make following operation as many times as you wish (possibly zero or more). In a single operation, you can swap any two entries of the matrix. Your aim is to minimize the bandwidth of the matrix. Find the minimum bandwidth of the matrix A you can get after making as many operations of above type as you want.

-----Input-----
The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follow.
First line of each test case contains an integer N denoting the height/width of the matrix.
Next N lines of each test case contain N space separated binary integers (either zero or one) corresponding to the entries of the matrix.

-----Output-----
For each test case, output a single integer corresponding to the minimum bandwidth that you can obtain.

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ N ≤ 500
- 0 ≤ A(i, j) ≤ 1

-----Subtasks-----
- Subtask #1 (40 points) : 1 ≤ N ≤ 100
- Subtask #2 (60 points) : original constraints

-----Example-----
Input:
6
2
0 0
0 0
2
1 0
0 1
2
1 0
1 0
2
1 0
1 1
3
1 0 0
0 1 1
1 1 0
4
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1

Output:
0
0
0
1
1
3

-----Explanation-----
Example case 1. The bandwidth of a matrix will all zero entries will be zero. This is the minimum bandwidth you can get, so there is no need of performing any swap operation.
Example case 2. The bandwidth of a diagonal matrix will also be zero.
Example case 3. You can make the given matrix a diagonal matrix by swapping A(2, 1) and A(2, 2), which will have zero bandwidth.
Example case 4. You can not make swaps in any way that can reduce the bandwidth of this matrix. Bandwidth of this matrix is equal to 1, which is the minimum bandwidth that you can get.
Example case 5. Bandwidth of the given matrix is 2. You can make it equal to be 1 by swapping A(3, 1) and A(3, 3), i.e. the matrix after the operation will look like

1 0 0
0 1 1
0 1 1

The bandwidth of this matrix is 1.

Example case 6. The swap operations won't have any effect on the matrix. Its bandwidth is equal to 3.
user: There's a tree and every one of its nodes has a cost associated with it. Some of these nodes are labelled special nodes. You are supposed to answer a few queries on this tree. In each query, a source and destination node (SNODE$SNODE$ and DNODE$DNODE$) is given along with a value W$W$. For a walk between SNODE$SNODE$ and DNODE$DNODE$ to be valid you have to choose a special node and call it the pivot P$P$. Now the path will be SNODE$SNODE$ ->P$ P$ -> DNODE$DNODE$. For any valid path, there is a path value (PV$PV$) attached to it. It is defined as follows:
Select a subset of nodes(can be empty)  in the path from SNODE$SNODE$ to P$P$ (both inclusive) such that sum of their costs (CTOT1$CTOT_{1}$) doesn't exceed W$W$.
Select a subset of nodes(can be empty) in the path from P$P$ to DNODE$DNODE$ (both inclusive) such that sum of their costs (CTOT2$CTOT_{2}$) doesn't exceed W$W$.
Now define PV=CTOT1+CTOT2$PV = CTOT_{1} + CTOT_{2}$ such that the absolute difference  x=|CTOT1−CTOT2|$x = |CTOT_{1} - CTOT_{2}|$ is as low as possible. If there are multiple pairs of subsets that give the same minimum absolute difference, the pair of subsets which maximize PV$PV$ should be chosen.
For each query, output the path value PV$PV$ minimizing x$x$ as defined above. 
Note that the sum of costs of an empty subset is zero.

-----Input-----
- First line contains three integers N$N$ - number of vertices in the tree, NSP$NSP$ - number of special nodes in the tree and Q$Q$ - number of queries to answer.   
- Second line contains N−1$N-1$ integers. If the i$i$th integer is Vi$V_i$ then there is an undirected edge between i+1$i + 1$ and Vi$V_i$ (i$i$ starts from 1$1$ and goes till N−1$N-1$).          
- Third line contains N$N$ integers, the i$i$th integer represents cost of the i$i$th vertex.    
- Fourth line contains NSP$NSP$ integers - these represent which nodes are the special nodes.    
- Following Q$Q$ lines contains three integers each - SNODE$SNODE$, DNODE$DNODE$ and W$W$ for each query.

-----Output-----
For each query output a single line containing a single integer - the path value PV$PV$ between SNODE$SNODE$ and DNODE$DNODE$.

-----Constraints:-----
- 1≤$1 \leq $ Number of nodes ≤1000$ \leq 1000 $ 
- 0≤W≤1000$ 0 \leq W \leq 1000 $ 
- 1≤$ 1 \leq $ Number of special nodes ≤10$ \leq 10 $ 
- 0≤$ 0 \leq $ Cost of each node ≤1000$ \leq 1000 $ 
- 1≤$ 1 \leq $ Number of queries ≤1000$ \leq 1000 $

-----Sample Input-----
7 1 5

1 1 2 2 3 3

3 5 4 2 7 9 1

1

2 3 100

1 1 100

2 1 100

4 5 100

4 7 100      

-----Sample Output:-----
6

6

6

20

16       

-----Explanation:-----
Consider query 4$4$. The only path is 4−>2−>1−>2−>5$4->2->1->2->5$. The two sets defined for this path are {3,2,5${3,2,5}$} and {3,5,7${3,5,7}$}. Pick subsets {3,2,5${3,2,5}$} and {3,7${3,7}$} from each set which minimizes PV$PV$. Note that node 2$2$ can repeat as it is in different paths (both to and from the pivot).
user: Indraneel's student has given him data from two sets of experiments that the student has performed. Indraneel wants to establish a correlation between the two sets of data. Each data set is a sequence of $N$ numbers. The two data sets do not match number for number, but Indraneel believes that this is because data has been shifted due to inexact tuning of the equipment.
For example, consider the following two sequences:
$ $
3  8   4  23  9  11  28
2  3  22  26  8  16  12

$ $
Indraneel observes that if we consider the subsequences $3,4,23,9$ and $2,3,22,8$ and examine their successive differences we get $1,19,-14$. He considers these two subsequences to be "identical". He would like to find the longest such pair of subsequences so that the successive differences are identical. Your task is to help him do this.

-----Input:-----
The first line of the input will contain a single integer $N$ indicating the number of data points in each of Indraneel's student's data sets. This is followed by two lines, each containing $N$ integers.

-----Output:-----
The output consists of three lines. The first line of output contains a single integer indicating the length of the longest pair of subsequences (one from each sequence) that has identical successive differences. This is followed by two lines each containing the corresponding subsequences. If there is more than one answer, it suffices to print one.

-----Constraints:-----
- $1 \leq N \leq 150$.
- $0 \leq$ Each data point $\leq 1000$

-----Sample Input-----
7
3 8 4 23 9 11 28  
2 3 22 26 8 16 12 

-----Sample Output-----
4
3 4 23 9
2 3 22 8
user: Harrenhal is the largest castle in the Seven Kingdoms and is the seat of House Whent in the Riverlands, on the north shore of the Gods Eye lake. Since the War of Conquest, however, it has become a dark and ruinous place.
(c) A Wiki of Ice and Fire

Now Harrenhal is too dangerous since it's a nice place for bandits to hide, or even for rebels to start planning overthrowing of the king. So, the current Lord of the Seven Kingdoms has decided, that it's time to completely ruin the castle. For that puposes, he's planning to send some military troops.
In this problem we assume, that Harrenhal can be described as a string H, which consists only of symbols 'a' and 'b'. Harrenhal is completely ruined if and only if the length of H is equal to zero.
So, how to make H empty? Send a military troop! When a military troop of the king reach the castle, they delete some palindromic subsequence S of H. For example, let H = 'abbabaab'. Then the current military troop can choose S = 'ababa'(Let's make symbols of S bold in H: 'abbabaab'). After deleting S, H will be equal to 'bab'. Military troops are free to choose any possible palindromic subsequence of H.
Your task is pretty simple: determine the minimal number of military troops, that the Lord of the Seven Kingdoms has to send in order to ruin Harrenhal.

-----Note-----

Maybe, some of you aren't familiar with definitions from the statement. Here're some articles that could help you to understand the problem correctly:

- Subsequence: http://en.wikipedia.org/wiki/Subsequence
- Palindrome: http://en.wikipedia.org/wiki/Palindrome

-----Input-----
The first line of the input contains an integer T, denoting the number of test cases.
The next T lines contain a string H each, denoting the string, that describes the current state of Harrenhal for the corresponding test cases.
It's guaranteed, that each H consists only of symbols 'a' and 'b'.

-----Output-----
The output should contain exactly T lines. i'th line of the output should contain the only integer: the minimal number of military troops, that the Lord of the Seven Kingdoms has to send in order to ruin Harrenhal for the corresponding test cases.

-----Constraints-----
- 1 ≤ |H| ≤ 100000, for each H.
- Subtask 1(30 points): each H in the input is a palindrome, 1 ≤ T ≤ 6;
- Subtask 2(70 points): 1 ≤ T ≤ 9.

-----Example-----
Input:
1
abbabaab

Output:
2

-----Explanation-----
There're multiple ways to ruin Harrenhal in the example test. Let's consider one of them.
The first troop can delete S = 'ababa'('abbabaab'). After that, H = 'bab'.
The second troop can delete S = 'bab'('bab'). After that, H is empty and that's it.
user: A prime number is number x which has only divisors as 1 and x itself.
Harsh is playing a game with his friends, where his friends give him a few numbers claiming that they are divisors of some number x but divisor 1 and the number x itself are not being given as divisors.
You need to help harsh find which number's divisors are given here.
His friends can also give him wrong set of divisors as a trick question for which no number exists.
Simply, We are given the divisors of a number x ( divisors except 1 and x itself ) , you have to print the number if only it is possible.
You have to answer t queries.
(USE LONG LONG TO PREVENT OVERFLOW)

-----Input:-----
- First line is T queires.
- Next are T queries.
- First line is N ( No of divisors except 1 and the number itself )
- Next line has N integers or basically the divisors.

-----Output:-----
Print the minimum possible x which has such divisors and print -1 if not possible.

-----Constraints-----
- 1<= T <= 30
- 1<= N <= 350
- 2<= Di <=10^6

-----Sample Input:-----
3
2
2 3
2
4 2
3
12 3 2

-----Sample Output:-----
6
8
-1

-----EXPLANATION:-----
Query 1 : Divisors of 6 are ( 1,2,3,6) Therefore, Divisors except 1 and the number 6 itself are ( 2 , 3). Thus, ans = 6.
Query 2 : Divisors of 8 are ( 1,2,4,8) Therefore, Divisors except 1 and the number 8 itself are ( 2 , 4). Thus, ans = 8.
Query 3 : There is no such number x with only ( 1,2,3,12,x ) as the divisors.
user: Mandarin chinese
, Russian and Vietnamese as well.
Let's denote $S(x)$ by the sum of prime numbers that divides $x$.
You are given an array $a_1, a_2, \ldots, a_n$ of $n$ numbers, find the number of pairs $i, j$ such that $i \neq j$, $a_i$ divides $a_j$ and $S(a_i)$ divides $S(a_j)$.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- First line of each testcase contains one integer $n$ — number of elements of the array.
- Second line of each testcase contains $n$ space-separated integers $a_1, a_2, \ldots, a_n$.

-----Output:-----
For each testcase, output in a single line number of pairs that each of it satisfies given conditions.

-----Constraints-----
- $1 \leq T \leq 100$
- $2 \leq n, a_i \leq 10^6$
- the sum of $n$ for all test cases does not exceed $10^6$

-----Subtasks-----
Subtask #2 (20 points): $2 \leq n \leq 100$, $2 \leq a_i \leq 10^4$
Subtask #2 (80 points): original contsraints

-----Sample Input:-----
1
5
2 30 2 4 3

-----Sample Output:-----
6

-----EXPLANATION:-----
$S(2) = 2, S(30) = 2 + 3 + 5 = 10, S(4) = 2, S(3) = 3$. So using this information, the pairs of indicies are $(1,2)$, $(1, 3)$, $(1, 4)$, $(3, 1)$, $(3, 2)$, $(3, 4)$.
user: Chef has arrived in Dagobah to meet with Yoda to study cooking. Yoda is a very busy cook and he doesn't want to spend time with losers. So he challenges the Chef to a series of games, and agrees to teach the Chef if Chef can win at least P of the games. The total number of games is K. The games will be played on a chessboard of size N*M. That is, it has N rows, each of which has M squares. At the beginning of the game, a coin is on square (1, 1), which corresponds to the top-left corner, and every other square is empty. At every step, Yoda and Chef have to move the coin on the chessboard. The player who cannot make a move loses. Chef makes the first move. They can't move the coin to a square where it had been placed sometime before in the game, and they can't move outside chessboard.

In this game, there are two different sets of rules according to which the game can be played:

-from (x, y) player can move coin to (x+1, y), (x-1, y), (x, y+1), (x, y-1) in his turn, if they are valid squares.

-from (x, y) player can move coin to (x+1, y+1), (x-1, y-1), (x-1, y+1), (x+1, y-1) in his turn, if they are valid squares.

Before every game, the Power of the kitchen chooses one among the two sets of rules with equal probability of 0.5, and the game will be played according to those rules. Chef and Yoda are very wise, therefore they play optimally. You have to calculate the probability that Yoda will teach Chef.

-----Input-----

Input begins with an integer T, the number of test cases

Each test case begins with 4 integers N, M, P, K.

-----Output-----

For each test case, output a line containing the answer for task. The output must have an absolute error at most 0.000001 (10-6).  

-----Constraints and Subtasks-----
-  1 ≤ T ≤ 50 
-  1 ≤ K 

Subtask 1 : 10 points 

-  2 ≤ N, M ≤ 5 
-  0 ≤ P ≤ K ≤ 5 

Subtusk 2 : 10 points 
-  2 ≤ N, M ≤ 10 
-  0 ≤ P ≤ K ≤ 10^3 

Subtusk 3 : 20 points 
-  2 ≤ N, M ≤ 100 
-  0 ≤ P ≤ K ≤ 10^3 

Subtusk 4 : 60 points
-  2 ≤ N, M ≤ 100 
-  0 ≤ P ≤ K ≤ 10^6 

-----Example-----
Input:
2
2 3 2 3
2 2 5 5
Output:
0.500000
1.000000
user: You are given an array $A$ of $N$ positive and pairwise distinct integers. 
You can permute the elements in any way you want. 
The cost of an ordering $(A_1, A_2, \ldots, A_N)$ is defined as $ (((A_1 \bmod A_2) \bmod A_3)......) \bmod A_N$ where $X \bmod Y$ means the remainder when $X$ is divided by $Y$. 
You need to find the maximum cost which can be attained through any possible ordering of the elements.

-----Input:-----
- The first line contains $T$ denoting the number of test cases.
- The first line of each testcase contains a single integer $N$. 
- The second line of each testcase contains $N$ space-separated integers, the elements of $A$. 

-----Output:-----
- For each testcase, output the maximum possible cost in a new line. 

-----Constraints-----
- $1 \leq T \leq 5*10^5$
- $2 \leq N \leq 5*10^5$
- $1 \leq A_i  \leq 10^9$
- Sum of $N$ over all testcases is less than or equal to $10^6$
- All elements in a single testcase are distinct.

-----Subtasks-----
- 100 points : Original constraints.

-----Sample Input:-----
1
2
7 12

-----Sample Output:-----
7

-----Explanation:-----
The two possible ways to order the elements are [7, 12] and [12, 7]. In the first case, the cost is $7 \bmod 12 = 7$ and in the second case the cost is $12 \bmod 7 = 5$. Clearly the answer is 7.
user: Chef is again playing a game with his best friend Garry. As usual, the rules of this game are extremely strange and uncommon.
First, they are given a stack of $N$ discs. Each disc has a distinct, non-negative integer written on it. The players exchange turns to make a move. Before the start of the game, they both agree upon a set of positive integers  $S$ of size $K$. It is guaranteed that S contains the integer $1$. In a move, a player can select any value $x$ from $S$ and pop exactly $x$ elements from the top of the stack. The game ends when there are no discs remaining. Chef goes first.
Scoring: For every disc a player pops, his score increases by $2^p$ where $p$ is the integer written on the disc. For example, if a player pops the discs, with integers $p_1, p_2, p_3, \dots, p_m$ written on it, during the entire course of the game, then his total score will be $2^{p_1} + 2^{p_2} + 2^{p_3} + \dots + 2^{p_m}$.
The player with higher score wins the game. Determine the winner if both the players play optimally, or if the game ends in a draw.

-----Input:-----
- First line contains $T$, the number of testcases. Then the testcases follow.
- The first line of each test case contains two space separated integers $N$ and $K$, denoting the size of the stack and the set S respectively.
- Next line contains $N$ space separated integers $A_i$ where $A_1$ is the topmost element, denoting the initial arrangement of the stack.
- The last line of each test case contains $K$ space separated integers each denoting $x_i$.

-----Output:-----
For each testcase, output "Chef" (without quotes) if Chef wins, "Garry" (without quotes) if Garry wins, otherwise "Draw" (without quotes) in a separate line.

-----Constraints-----
- $1 \leq T \leq 1000$
- $1 \leq N \leq 10^5$
- $1 \leq K \leq \min(100, N)$
- $0 \leq A_i \leq 10^9$
- $1 \leq x_i \leq N$
- $x_i \neq x_j$ for all $i \neq j$
- $A_i \neq A_j$ for all $i \neq j$
- Set $S$ contains integer $1$. 
- Sum of $N$ over all test cases does not exceed $10^5$.

-----Sample Input:-----
1
3 2
5 7 1
1 2

-----Sample Output:-----
Chef

-----Explanation:-----
Chef can select 2 from the set and draw the top two discs (with integers 5 and 7 written on it) from the stack. Garry cannot select 2 from the set as there is only 1 disc left in the stack. However, he can select 1 from the set and pop the last disc.
So, Chef's score = $2^5$ + $2^7$ = $160$

Garry's score = $2^1$ = $2$

Chef wins.
user: Chef is baking a cake.

While baking, in each minute the size of cake doubles as compared to its previous size.

In this cake, baking of cake is directly proportional to its size.
You are given $a$, the total time taken(in minutes) to bake the whole cake.

Let cake be half baked at $k^{th}$ minute.         
Your task is to find the value of $k+2$.

-----Input:-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of T test cases follows.               
- The first and only line of each test case contains a single integer $a$.

-----Output:-----
For each testcase , print one line, the value of $k+2$.

-----Constraints-----
- $1 \leq T \leq 8 $
- $2 \leq a \leq 10^{128}$

-----Sample Input:-----
1

2

-----Sample Output:-----
3

-----Explaination-----
Time was 1 min when cake was half baked by chef so answer is 1+2=3
user: Once N boys and M girls attended a party. You are given a matrix A of N rows and M columns where Aij is 1 if the i-th boy likes the j-th girl, otherwise it will be 0. Note that it is not necessary that if a boy x likes girl y, then girl y should like boy x.
You know that if there are two different boys x and y, who both like girl z, then there will be a collision.
Can you calculate the number of different collisions at this party? Note that order of boys in the collision doesn't matter.

-----Input-----
The first line contains a single integer T denoting the number of test cases. Then T test cases follow.
The first line of each test case contains two space separated integers N, M denoting the number of boys and girls, respectively.
Each of the following N lines contain M characters, each of them is either '0' or '1'.

-----Output-----
For each test case output a single line containing an integer corresponding to the number of collisions at the party.

-----Constraints-----
- 1 ≤ T ≤ 100
- 1 ≤ N, M ≤ 10

-----Example-----
Input:
2
4 3
111
100
110
000
2 2
10
01

Output:
4
0

-----Explanation-----
Example Case 1. All three boys like the first girl, so there are (1, 2, 1), (1, 3, 1), (2, 3, 1) collisions with her. Boys 1 and 3 both like the second girl so this is one more collision. Only one boy likes the third girl, so there are no collisions with her and thus we have 4 collisions total.
Example Case 2. For each girl there is only one boy who likes her, so there are no collisions at all.