meetforever/GRPO_CL huggingface dataset:
user: Chef is stuck in a two dimensional maze having N rows and M columns. He needs to get out of the maze as soon as possible and arrive at the kitchen in order to serve his hungry customers. But, he can get out of the maze only if he is able to successfully find any magical path in the given maze.

A path is defined as magical if it starts from any of the cell (a,b) of the maze and ends at the cell (c,d) such that the following conditions are satisfied :-
- |a - c| + |b - d| = 1
- All the cells in the maze are traversed exactly once.
- It is allowed to move only in the four directions - up, down, left and right from the current cell.

-----Input-----
- First line of the input contains an integer T denoting the number of different types of scenarios.
- Each of the next T lines will contain two integers N, M denoting the dimensions of the maze.

-----Output-----
For each of the T scenarios, output a single line containing "Yes" or "No" (without quotes) denoting whether the Chef can get out of the maze or not.

-----Constraints-----
- 1 ≤ T ≤ 105
- 1 ≤ N, M ≤ 1018

-----Subtasks-----
Subtask #1 : (30 points)
- 1 ≤ T ≤ 100
- 1 ≤ N, M ≤ 10

Subtask #2 : (70 points) 

Original Constraints

-----Example-----
Input:
1
2 2

Output:
Yes

-----Explanation-----
Example case 1.

Chef can start from (1,1), move down to (2,1), then move right to (2,2) and finally move upwards to reach (1,2). As, he is able to visit all the cells exactly once and sum of absolute differences of corresponding x and y dimension is 1, we can call this path a magical path.
user: Little Egor is a huge movie fan. He likes watching different kinds of movies: from drama movies to comedy movies, from teen movies to horror movies. He is planning to visit cinema this weekend, but he's not sure which movie he should watch.
There are n movies to watch during this weekend. Each movie can be characterized by two integers Li and Ri, denoting the length and the rating of the corresponding movie. Egor wants to watch exactly one movie with the maximal value of Li × Ri. If there are several such movies, he would pick a one with the maximal Ri among them. If there is still a tie, he would pick the one with the minimal index among them.
Your task is to help Egor to pick a movie to watch during this weekend.

-----Input-----
The first line of the input contains an integer T denoting the number of test cases.
The first line of the test case description contains an integer n.
The second line of the test case description contains n integers L1, L2, ...,Ln. The following line contains n integers R1, R2, ..., Rn.

-----Output-----
For each test case, output a single integer i denoting the index of the movie that Egor should watch during this weekend. Note that we follow 1-based indexing.

-----Constraints-----
- 1 ≤ T ≤ 5
- 1 ≤ n ≤ 100
- 1 ≤ Li, Ri ≤ 100

-----Example-----
Input:
2
2
1 2
2 1
4
2 1 4 1
2 4 1 4

Output:
1
2

-----Explanation-----
In the first example case, both films have the same value of L × R, but the first film has a better rating.
In the second example case, the second and the fourth movies are equally good, but the second movie has a smaller index.
user: Find sum of all the numbers that are multiples of 10 and are less than or equal to a given number "N". (quotes for clarity and be careful of integer overflow)

-----Input-----
Input will start with an integer T the count of test cases, each case will have an integer N.

-----Output-----
Output each values, on a newline.

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ N ≤1000000000

-----Example-----
Input:
1
10

Output:
10

-----Explanation-----
Example case 1. Only integer that is multiple 10 that is less than or equal to 10 is 10
user: Chef and his girlfriend are going to have a promenade. They are walking along the straight road which consists of segments placed one by one. Before walking Chef and his girlfriend stay at the beginning of the first segment, they want to achieve the end of the last segment. 
There are few problems: 

-  At the beginning Chef should choose constant integer - the velocity of mooving. It can't be changed inside one segment. 
-  The velocity should be decreased by at least 1 after achieving the end of some segment. 
-  There is exactly one shop on each segment. Each shop has an attractiveness. If it's attractiveness is W and Chef and his girlfriend move with velocity V then if V < W girlfriend will run away into the shop and the promenade will become ruined. 

Chef doesn't want to lose her girl in such a way, but he is an old one, so you should find the minimal possible velocity at the first segment to satisfy all conditions.


-----Input-----
- The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.
- The first line of each test case contains a single integer N denoting the number of segments. The second line contains N space-separated integers W1, W2, ..., WN denoting the attractiveness of shops. 

-----Output-----
- For each test case, output a single line containing the minimal possible velocity at the beginning.

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ N ≤ 10^5
- 1 ≤ Wi ≤ 10^6

-----Example-----
Input:

2
5
6 5 4 3 2
5
3 4 3 1 1

Output:

6
5

-----Explanation-----
Example case 1. 
If we choose velocity 6, on the first step we have 6 >= 6 everything is OK, then we should decrease the velocity to 5 and on the 2nd segment we'll receive 5 >= 5, again OK, and so on. 
Example case 2. 
If we choose velocity 4, the promanade will be ruined on the 2nd step (we sould decrease our velocity, so the maximal possible will be 3 which is less than 4).
user: Dhiraj loves Chocolates.He loves chocolates so much that he can eat up to $1000$ chocolates a day. But his mom is fed up by this habit of him and decides to take things in her hand.
Its diwali Season and Dhiraj has got a lot of boxes of chocolates and Dhiraj's mom is afraid that dhiraj might eat all boxes of chocolates.
So she told Dhiraj that he can eat only exactly $k$ number of chocolates and dhiraj has to finish all the chocolates in box selected by him and then move on to next box of chocolate.Now Dhiraj is confused that whether he will be able to eat $k$ number of chocolates or not. Since dhiraj is weak at maths,he asks for your help to tell him whether he can eat $k$ number of chocolates or not. 
So given number of chocolates  are $k$ which dhiraj has to eat and the boxes of chocolates each containing some number of chocolates, tell whether dhiraj will be able to eat $k$ number of chocolates or not.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- $k$, representing the number of chocolates dhiraj has to eat.
- the third line contains $N$ representing the no. of boxes of chocolates.
- fourth line contains list of $a[]$ size $N$ specifying the number of chocolates in each Box.

-----Output:-----
- For each testcase, output in a single line answer $0$ or $1$.
- $0$ if dhiraj cant eat $k$ chocolates from given combination and $1$ if he can eat $k$ chocolates from given combination.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq K \leq 10^7$
- $1 \leq  N \leq 800$
- $1 \leq a[i] \leq 10^3$

-----Sample Input:-----
2
20
5
8 7 2 10 5
11
4
6 8 2 10

-----Sample Output:-----
1
0
user: The chef is trying to solve some pattern problems, Chef wants your help to code it. Chef has one number K(odd) to form a new pattern. Help the chef to code this pattern problem.

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains a single line of input, one integer $K$. 

-----Output:-----
For each test case, output as the pattern.

-----Constraints-----
- $1 \leq T \leq 50$
- $1 \leq K \leq 50$

-----Sample Input:-----
4
1
3
5
7

-----Sample Output:-----
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: The Petrozavodsk camp takes place in about one month. Jafar wants to participate in the camp, but guess what? His coach is Yalalovichik.
Yalalovichik is a legendary coach, famous in the history of competitive programming. However, he is only willing to send to the camp students who solve really hard problems on Timus. The deadline that Yalalovichik set before has passed and he refuses to send Jafar to the camp.
Jafar decided to make Yalalovichik happy in hopes of changing his decision, so he invented a new sequence of numbers and named them Yalalovichik numbers. Jafar is writing a research paper about their properties and wants to publish it in the Science Eagle yearly journal.
A Yalalovichik number is created in the following way:
- Consider an integer $N$ in decimal notation; let's call it the base of the Yalalovichik number $Y_N$. $N$ may not contain the digit $0$.
- Treat $N$ as a decimal string. Compute all left shifts of this string $N_0, N_1, \ldots, N_{|N|-1}$ ($|N|$ denotes the number of digits of $N$); specifically, $N_k$ denotes the string formed by moving the first $k$ digits of $N$ to the end in the same order.
- Concatenate the strings $N_0, N_1, \ldots, N_{|N|-1}$. The resulting string is the decimal notation of $Y_N$.
For example, if $N = 123$, the left shifts are $123, 231, 312$ and thus $Y_N = 123231312$.
You are given the base $N$. Calculate the value of $Y_N$ modulo $10^9+7$.

-----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 decimal integer $N$.

-----Output-----
For each test case, print a single line containing one integer — the value of the Yalalovichik number $Y_N$ modulo $10^9+7$.

-----Constraints-----
- $1 \le T \le 200$
- $|N| \le 10^5$
- $N$ does not contain the digit $0$
- the sum of $|N|$ over all test cases does not exceed $10^6$

-----Example Input-----
1
123

-----Example Output-----
123231312
user: The chef is trying to solve some pattern problems, Chef wants your help to code it. Chef has one number K to form a new pattern. Help the chef to code this pattern problem.

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains a single line of input, one integer $K$. 

-----Output:-----
For each test case, output as the pattern.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq K \leq 100$

-----Sample Input:-----
3
5
3
4

-----Sample Output:-----
1   1
2 2
3
4 4
5   5
1 1
2
3 3
1  1
22
33
4  4

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: You are given N integer sequences A1, A2, ..., AN. Each of these sequences contains N elements. You should pick N elements, one from each sequence; let's denote the element picked from sequence Ai by Ei. For each i (2 ≤ i ≤ N), Ei should be strictly greater than Ei-1.
Compute the maximum possible value of E1 + E2 + ... + EN. If it's impossible to pick the elements E1, E2, ..., EN, print -1 instead.

-----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 line of each test case contains a single integer N.
- N lines follow. For each valid i, the i-th of these lines contains N space-separated integers Ai1, Ai2, ..., AiN denoting the elements of the sequence Ai.

-----Output-----
For each test case, print a single line containing one integer — the maximum sum of picked elements.

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ N ≤ 700
- 1 ≤ sum of N in all test-cases ≤ 3700
- 1 ≤ Aij ≤ 109 for each valid i, j

-----Subtasks-----
Subtask #1 (18 points): 1 ≤ Aij ≤ N for each valid i, j
Subtask #2 (82 points): original constraints

-----Example-----
Input:

1
3
1 2 3
4 5 6
7 8 9

Output:

18

-----Explanation-----
Example case 1: To maximise the score, pick 3 from the first row, 6 from the second row and 9 from the third row. The resulting sum is E1+E2+E3 = 3+6+9 = 18.
user: Little kids, Jack and Evan like playing their favorite game Glass-and-Stone. Today they want to play something new and came across Twitter on their father's laptop.

They saw it for the first time but were already getting bored to see a bunch of sentences having at most 140 characters each. The only thing they liked to play with it is, closing and opening tweets.

There are N tweets on the page and each tweet can be opened by clicking on it, to see some statistics related to that tweet. Initially all the tweets are closed. Clicking on an open tweet closes it and clicking on a closed tweet opens it. There is also a button to close all the open tweets. Given a sequence of K clicks by Jack, Evan has to guess the total number of open tweets just after each click. Please help Evan in this game.

-----Input-----
First line contains two integers N K, the number of tweets (numbered 1 to N) and the number of clicks respectively (1 ≤ N, K ≤ 1000). Each of the following K lines has one of the following.

- CLICK X , where X is the tweet number (1 ≤ X ≤ N)
- CLOSEALL

-----Output-----
Output K lines, where the ith line should contain the number of open tweets just after the ith click.

-----Example-----
Input:
3 6
CLICK 1
CLICK 2
CLICK 3
CLICK 2
CLOSEALL
CLICK 1

Output:
1
2
3
2
0
1

Explanation:

Let open[x] = 1 if the xth tweet is open and 0 if its closed.

Initially open[1..3] = { 0 , 0 , 0 }. Here is the state of open[1..3] after each click and corresponding count of open tweets.

CLICK 1 : { 1, 0, 0 }, open count = 1

CLICK 2 : { 1, 1, 0 }, open count = 2

CLICK 3 : { 1, 1, 1 }, open count = 3

CLICK 2 : { 1, 0, 1 }, open count = 2

CLOSEALL : { 0, 0, 0 }, open count = 0

CLICK 1 : { 1, 0, 0 }, open count = 1
user: Give me Biscuit

Sunny wants to make slices of biscuit of size c * d into identical pieces.

but each piece is a square having maximum possible side length with no left over piece of biscuit.

Input Format

The first line contains an integer N.

N lines follow. Each line contains two space separated integers c and d.

which denote length and breadth of the biscuit.

Constraints

1 <= N <= 1000

1 <= c,d <= 1000
Output Format

N lines, each containing an integer that denotes the number of squares of maximum size, when the biscuit is cut as per the given condition.

Sample Input 

2

2 2

6 9

Sample Output 

1

6

Explanation 
	

The 1st testcase has a biscuit whose original dimensions are 2 X 2, the biscuit is uncut and is a square.

Hence the answer is 1.

The 2nd testcase has a biscuit of size 6 X 9 . We can cut it into 54 squares of size 1 X 1 , 6 of size 3 X 3 . For other sizes we will have leftovers.

Hence, the number of squares of maximum size that can be cut is 6.
user: -----Problem Statement-----
Harry Potter has one biscuit and zero rupee in his pocket. He will perform the following operations exactly $K$ times in total, in the order he likes:
- Hit his pocket, which magically increases the number of biscuits by one.
- Exchange $A$ biscuits to $1$ rupee.
- Exchange $1$ rupee to $B$ biscuits.
Find the maximum possible number of biscuits in Harry's pocket after $K$ operations.

-----Input-----
Input is given in the following format:
K A B

-----Output-----
Print the maximum possible number of biscuits in Harry's pocket after $K$operations.

-----Constraints-----
- $1 \leq K, A, B \leq 10^9$
- $K$, $A$ and  are integers.

-----Sample Input-----
4 2 6

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

-----EXPLANATION-----
The number of biscuits in Harry's pocket after $K$ operations is maximized as follows:
- Hit his pocket. Now he has $2$ biscuits and $0$ rupee.
- Exchange $2$ biscuits to $1$ rupee in his pocket .Now he has $0$ biscuits and $1$ rupee.
- Hit his pocket. Now he has $1$ biscuits and $1$ rupee.
- Exchange $1$ rupee to $6$ biscuits. his pocket. Now he has $7$ biscuits and $0$ rupee.
user: Did you know that the people of America eat around 100 acres of pizza per day ? Having read this fact on internet, two chefs from the Elephant city, Arjuna and Bhima are set to make pizza popular in India. They organized a social awareness camp, where N people ( other than these two ) sit around a large pizza. To make it more interesting, they make some pairs among these N people and the two persons in a pair, feed each other.

Each person should be a part of at most one pair and most importantly, to make feeding easy, any two pairs should not cross each other ( see figure for more clarity ). Arjuna and Bhima decided to play a game on making the pairs. In his turn, a player makes a pair ( selects two persons, as long as its valid ) and this pair start feeding each other. Arjuna and Bhima take turns alternately, by making a pair in each turn, and they  play optimally ( see Notes for more clarity ). The one who can not make a pair in his turn, loses. Given N, find who wins the game, if Arjuna starts first. 

-----Notes-----
- 'Optimally' means, if there is a possible move a person can take in his turn that can make him win finally, he will always take that. You can assume both are very intelligent. 

-----Input-----
First line contains an integer T ( number of test cases, around 1000 ). Each of the next T lines contains an integer N ( 2 <= N <= 10000 )

-----Output-----
For each test case, output the name of the winner ( either "Arjuna" or "Bhima" ( without quotes ) ) in a new line.

-----Example-----
Input:
4
2
4
5
6

Output:
Arjuna
Arjuna
Bhima
Arjuna

Explanation:

Let the people around the table are numbered 1, 2, ... , N in clock-wise order as shown in the image 

Case 1 : N = 2. Only two persons and Arjuna makes the only possible pair (1,2)

Case 2 : N = 4. Arjuna can make the pair (1,3). Bhima can not make any more pairs ( without crossing the pair (1,3) )

Case 3 : N = 5. No matter which pair Arjuna makes first, Bhima can always make one more pair, and Arjuna can not make any further
user: Coach Moony wants the best team to represent their college in ICPC. He has $N$ students standing in a circle with certain rating $X_i$ on a competitive coding platform. It is an established fact that any coder with more rating on the platform is a better coder.      
Moony  wants to send his best $3$ coders based upon their rating. But all coders only want to have their friends in their team and every coder is friends with four other coders, adjacent two on left side in the circle, and adjacent two on right. So Moony comes up with a solution that team with maximum cumulative rating of all three members in a team shall be representing their college.
You need to give the cumulative score of the team that will be representing the college.

-----Input:-----
- First line will contain $T$, number of testcases. 
- First line of each test case contains a single integer $N$.
- Second  line of each test case takes $N$ integers, denoting rating of $ith$ coder.

-----Output:-----
For each testcase, output a single integer denoting cumulative rating  of the team.

-----Constraints-----
- $1 \leq T \leq 10$
- $7 \leq N \leq 10^5$ 
- $0 \leq X_i \leq 10^9$

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

7

10 40 30 30 20 0 0  

-----Sample Output:-----
100
user: We all know how great ABD aka AB-DE-VILLIERS is. However his team mates were jealous of him and posed a problem for him to solve.The problem description is as follows :

Given an array of integers,find the length of the largest subarray(contiguous) of the given array with the maximum possible GCD (Greatest Common Divisor).

For info on GCD ,see this link: https://en.wikipedia.org/wiki/Greatest_common_divisor

GCD of the subarray is defined as the GCD of all the elements of the subarray.
As ABD is not aware of competitive programming he asks your help. Help him!

-----Input-----
First line will contain integer N denoting the size of array.

Second line will contain N integers denoting array elements.

-----Output-----
The answer as specified in the problem statement .

-----Constraints-----
1 <= N <= 1000000

1 <= array[i] <=100000000000

-----Example-----
Input:
4
2 4 8 3

Output:
1

Explanation
GCD of all possible subarrays of the given array are : 2 , 2 , 2 , 1 , 4 , 4, 1 , 8 , 1 , 3

Largest GCD possible : 8

Length of the largest subarray with GCD as 8 is 1

Hence answer is 1 .
user: Chef has a sequence of positive integers $A_1, A_2, \ldots, A_N$. He wants to split this sequence into two non-empty (not necessarily contiguous) subsequences $B$ and $C$ such that $\mathrm{GCD}\,(B) + \mathrm{GCD}\,(C)$ is maximum possible. Help him find this maximum value.
Note: The greatest common divisor (GCD) of a sequence of positive integers is the largest positive integer that divides each element of this sequence. For example, the GCD of the sequence $(8, 12)$ is $4$.

-----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 line of each test case contains a single integer $N$.
- The second line contains $N$ space-separated integers $A_1, A_2, \ldots, A_N$.

-----Output-----
For each test case, print a single line containing one integer — the maximum value of $\mathrm{GCD}\,(B) + \mathrm{GCD}\,(C)$.

-----Constraints-----
- $1 \le T \le 10$
- $2 \le N \le 10^5$
- $1 \le A_i \le 10^9$ for each valid $i$

-----Subtasks-----
Subtask #1 (20 points): $2 \le N \le 20$
Subtask #2 (80 points): original constraints

-----Example Input-----
1              
4                    
4 4 7 6                    

-----Example Output-----
9

-----Explanation-----
Example case 1: For example, the sequence $A$ can be divided into subsequences $B = (4, 4, 6)$ and $C = (7)$.
user: Mr. X has come up with a new string compression algorithm. Consider a string of length N which contains up to K distinct characters. The compression algorithm works as follows: Replace each maximal contiguous substring containing only one distinct character (repeated an arbitrary number of times) and replace it by 2 values: the character and the length of the substring.
For example, the string "aabbaaa" will be compressed to "a, 2, b, 2, a, 3". Thus the length of the compressed string is 6.

Since Mr. X is living in advanced times, the length of any integer is considered to be 1. For example, if a string is compressed to "a, 111, b, 13", then its length after compression is considered to be 4.

To test his algorithm, he needs to know the expected length of the compressed string for given N and K if the input string is randomly uniformly chosen from all possibilities. He wants to run this experiment multiple times for different N, K and needs your help.

-----Input-----
The first line of the input contains an integer T denoting the number of queries. The description of T test cases follows.
The first and only line of each test case contains two integers N and K denoting the number of letters in the input string and the maximum number of distinct characters that can be present in the string.

-----Output-----

For each test case, output a single line containing the expected length of the compressed string. 
Your answer will be considered correct if the absolute error is less than 10-2

-----Constraints-----
- 1 ≤ T ≤ 105
- 1 ≤ N, K ≤ 109

-----Example-----
Input:
2
3 1
3 2
Output:
2.0
4.0

-----Explanation-----Example case 1:

There is only one string: aaa with compressed string = a, 3. Therefore length = 2
Example case 2

Input strings:

"aaa": "a, 3". Length = 2

"aab": "a, 2, b, 1". Length = 4 

"aba": "a, 1, b, 1, a, 1". Length = 6

"abb": "a, 1, b, 2". Length = 4

"baa": "b, 1, a, 2". Length = 4

"bab": "b, 1, a, 1, b, 1". Length = 6

"bba": "b, 2, a, 1". Length = 4

"bbb": "b, 3". Length = 2

Expected value = (2+4+6+4+4+6+4+2)/8 = 32/8 = 4
user: The chef is trying to decode some pattern problems, Chef wants your help to code it. Chef has one number K to form a new pattern. Help the chef to code this pattern problem.

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains a single line of input, one integer $K$. 

-----Output:-----
For each test case, output as the pattern.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq K \leq 100$

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

-----Sample Output:-----
0
01
10
010
101
010
0101
1010
0101
1010

-----EXPLANATION:-----
No need, else pattern can be decode easily.