meetforever/GRPO_CL huggingface dataset:
user: You are given a square matrix $M$ with $N$ rows (numbered $1$ through $N$) and $N$ columns (numbered $1$ through $N$). Initially, all the elements of this matrix are equal to $A$. The matrix is broken down in $N$ steps (numbered $1$ through $N$); note that during this process, some elements of the matrix are simply marked as removed, but all elements are still indexed in the same way as in the original matrix. For each valid $i$, the $i$-th step consists of the following:
- Elements $M_{1, N-i+1}, M_{2, N-i+1}, \ldots, M_{i-1, N-i+1}$ are removed.
- Elements $M_{i, N-i+1}, M_{i, N-i+2}, \ldots, M_{i, N}$ are removed.
- Let's denote the product of all $2i-1$ elements removed in this step by $p_i$. Each of the remaining elements of the matrix (those which have not been removed yet) is multiplied by $p_i$.
Find the sum $p_1 + p_2 + p_3 + \ldots + p_N$. Since this number could be very large, compute it 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 two space-separated integers $N$ and $A$.

-----Output-----
For each test case, print a single line containing one integer ― the sum of products at each step modulo $10^9+7$.

-----Constraints-----
- $1 \le T \le 250$
- $1 \le N \le 10^5$
- $0 \le A \le 10^9$
- the sum of $N$ over all test cases does not exceed $10^5$

-----Example Input-----
1
3 2

-----Example Output-----
511620149

-----Explanation-----
Example case 1:
user: $Gogi$, $Tapu$ and $Sonu$ are the elite members of $Tapu$ $Sena$. $Gogi$ is always stoned and asks absurd questions, But this time he asked a question which seems to be very serious and interesting. $Tapu$ wants to solve this question to impress $Sonu$. He  gave an array of length N to $Tapu$, $Tapu$ can perform the following operations exactly once:
- Remove any subarray from the given array given the resulting array formed after the removal is non-empty.    
- Reverse the whole array.
Remember you can’t shuffle the elements of the array.
Tapu needs to find out the maximum possible GCD of all the numbers in the array after applying the given operations exactly once. Tapu is very weak at programming, he wants you to solve this problem so that he can impress $Sonu$.

-----Input:-----
- The first line contains $T$, the number of test cases.
- For each test case
-FIrst line contains $N$.
- Last line contains $N$ numbers of the array. 

-----Output:-----
A single integer in a new line, maximum possible GCD. 

-----Constraints-----
- $1 \leq T \leq 10^2$
- $1 \leq N \leq 10^4$
- $1 \leq a[i] \leq 10^9$

Summation of N for all testcases is less than $10^6$ 

-----Sample Input 1:-----
1
1
2

-----Sample Output 1:-----
2
user: Blob is a computer science student. He recently got an internship from Chef's enterprise. Along with the programming he has various other skills too like graphic designing, digital marketing and social media management. Looking at his skills Chef has provided him different tasks A[1…N] which have their own scores. Blog wants to maximize the  value of the expression A[d]-A[c]+A[b]-A[a] such that d>c>b>a.

Can you help him in this?

-----Input:-----
- The first line contain the integer N
- The second line contains N space separated integers representing A[1], A[2] … A[N]

-----Output:-----
The maximum score that is possible

-----Constraints-----
- $4 \leq N \leq 10^4$
- $0 \leq A[i] \leq 10^5$

-----Sample Input:-----
6

3 9 10 1 30 40

-----Sample Output:-----
46
user: Chef got into a fight with the evil Dr Doof. Dr Doof has decided to destroy all even numbers from the universe using his Evil-Destroy-inator. Chef has $N$ integers with him. To stop Doof, Chef has to find an odd number which is an integer multiple of all $N$ numbers that he has with him. Find if it is possible for Chef to prevent Dr Doof from destroying the even numbers.
Formally, given $N$ positive integers, find if there exists an odd number which is an integer multiple of all the given $N$ numbers. If yes, print "YES", otherwise "NO". You can print any letter in any case.

-----Input-----
- First line contains $T$, number of testcases. Each testcase consists of $2$ lines.
- The first line of each test case consists of a positive integer $N$, denoting the number of positive integers Chef has.
- The second line of each test case contains $N$ space separated integers $A_i$ each denoting an integer that Chef has with him.

-----Output-----
For every test case, if there exists such an odd number, print "YES" on a separate line, otherwise "NO". The judge is case insensitive. That means, your code can print any letter in any case ( "Yes", "yes" or "YES" are all accepted).

-----Constraints-----
- $1 \leq T \leq 10^3$
- $1 \leq N \leq 10^3$
- $1 \leq A_i \leq 10^3$

-----Sample Input-----
2
5
1 2 5 4 3
1
7

-----Sample Output-----
NO
YES

-----Explanation-----
For test $1$: There exists no odd number.
For test $2$: The possible odd numbers can be $7$, $21$, $49$, $315$, …
user: Chef has a sequence of $N$ integers, $A_1, A_2, ... , A_N$. He likes this sequence if it contains a  subsequence of $M$ integers, $B_1, B_2, ... , B_M$ within it.
A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.
You will be given a sequence of $N$ integers, $A_1, A_2, ..., A_N$ followed by another sequence of $M$ integers, $B_1, B_2, ..., B_M$. Given these, you have to tell whether Chef likes the sequence of $N$ integers($A_1, A_2, ..., A_N$) or not. 
Formally, output "Yes" if
$\exists idx_1, idx_2, ..., idx_M | 1 \le idx_1 < idx_2 < ... < idx_M \le N$ and $A_{idx_i} = B_i \forall i, 1 \le i \le M$
Otherwise output "No". Note that the quotes are for clarity.

-----Input-----
The first line contains a single integer, $T$.
$T$ test cases follow where each test case contains four lines:
- The first line of a test case contains a single integer $N$
- The second line of the test case contains $N$ space separated integers, $A_1, A_2, ..., A_N$
- The third line of the test case contains a single integer $M$.
- The fourth line contains $M$ space separated integers, $B_1, B_2, ..., B_M$
Symbols have usual meanings as described in the statement.

-----Output-----
For each test case, output a single line containing the output. Output is "Yes" if Chef likes the sequence $A$. Output is "No" if Chef dislikes the sequence $A$.

-----Constraints-----
- $1 \le T \le 100$
- $1 \le N \le 10^3$
- $1 \le M \le 10^3$
- $1 \le A_i, B_i \le 10^9$

-----Sample Input-----
3
6
1 2 3 4 5 6
3
2 3 4
6
22 5 6 33 1 4
2
4 15
4
1 3 4 2
2
1 2

-----Sample Output-----
Yes
No
Yes

-----Explanation:-----
In sample test case $1$, the sequence $1,2,3,4,5,6$ contains the subsequence $2, 3, 4$. The subsequence is present at indices $1, 2, 3$ of the original sequence.
Hence, $1,2,3,4,5,6$ is a sequence which Chef likes it. Therefore, we output "Yes".
In sample test case $2$, the subsequence $4, 15$ is not present in sequence $22, 5, 6, 33, 1, 4$. Hence, we output "No".
In sample test case $3$, the sequence $1, 3, 4, 2$ contains the subsequence $1, 2$. The subsequence is present at indices $0, 3$. Therefore, we output "Yes".
user: It's John's birthday; he has brought some candies in distribute among N of his friends. Being a good friend, he decided to distribute all of his candies equally among his friend and he don't want to keep any for himself. He left the task of distributing candies up to his friends, but his friends haven't done a great job in distributing them and now they all have unequal amount of candies.
He asked all of his friends to keep all the candies in their hands (A[1] to A[N]) so that he can redistribute the candies. Taking a candy away from someone makes them sad and makes the person who gets it happy. Thus, you do not want to do many operations on the number of candies.
Here each operation stands for, taking away 1 candy from someone and giving it to somebody else.
After applying the operations, he might not be able to distribute all the candies among all of your friends, and you might need to buy some more candies as well. Buying each candy will be counted as 1 operation as well. John is saving up for his birthday party in the evening and so he wants to spend the least amount of money on candies as possible.
Thus, he asks for your help. You have to find the minimum number of operations required to make the distributions equal.

-----Input:-----
- The first line consists of the number T representing the number of test cases.
- Each test case consists of 2 lines.
- The first line contains of the number N representing the number of friends.
- The second line contains space separated numbers (A[1] to A[N]) representing the number of candies each friend currently has.

-----Output:-----
For each test case, you much print the minimum number of operations.

-----Constraints-----
- $1 \leq T \leq 10$
- $1 \leq N \leq 10^5$
- $0 \leq A[i] \leq 10^9$

-----Sample Input:-----
3

4

7 8 6 4

6

7 3 10 2 3 8

1

10   

-----Sample Output:-----
4

10

0
user: Recently Rocky had participated in coding competition and he is sharing one of the problem with you which he was unable to solve. Help Rocky in solving the problem.
Suppose the alphabets are arranged in a row starting with index 0$0$ from AtoZ$A to Z$.
If in a coded language  A=27$A=27$ and AND=65$AND=65$.
Help Rocky to find a suitable formula for finding all the value for given test cases?
(All alphabets are in Upper case only).

-----Input:-----
First line of the input contains  string s$s$.

-----Output:-----
Output the possible integer values of the given string s$s$ according to the question . 

-----Constraints-----
- 1≤s≤100$1 \leq s \leq 100$

-----Sample Input:-----
A

AND   

-----Sample Output:-----
27

65
user: Chef has created a special dividing machine that supports the below given operations on an array of positive integers.
There are two operations that Chef implemented on the machine.
Type 0 Operation

Update(L,R):
	for i = L to R:
		a[i] = a[i] / LeastPrimeDivisor(a[i])

Type 1 Operation

Get(L,R):
	result = 1
	for i = L to R:
		result = max(result, LeastPrimeDivisor(a[i]))
	return result;

The function LeastPrimeDivisor(x) finds the smallest prime divisor of a number. If the number does not have any prime divisors, then it returns 1.
Chef has provided you an array of size N, on which you have to apply M operations using the special machine. Each operation will be one of the above given two types. Your task is to implement the special dividing machine operations designed by Chef. Chef finds this task quite easy using his machine, do you too?

-----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 two space-separated integers N, M, denoting the size of array A and the number of queries correspondingly.

The second line of each test case contains N space-separated integers A1, A2, ..., AN denoting the initial array for dividing machine.
Each of following M lines contain three space-separated integers  type, L, R - the type of operation (0 - Update operation, 1 - Get operation), and the arguments of function, respectively

-----Output-----
For each test case, output answer of each query of type 1 (Get query) separated by space. Each test case from the same file should start from the new line.

-----Constraints-----
- 1 ≤ T ≤ 100
- 1 ≤ Ai ≤ 106
- 1 ≤ L ≤ R ≤ N
- 0 ≤ type ≤ 1
-  Sum of M over all test cases in a single test file does not exceed 106

-----Subtasks-----
Subtask #1: (10 points) 
- 1 ≤ N, M ≤ 103

Subtask #2: (25 points)
- 1 ≤ N, M ≤ 105
-  Ai is a prime number. 

Subtask #3: (65 points)
- 1 ≤ N, M ≤ 105

-----Example-----
Input:
2
6 7
2 5 8 10 3 44
1 2 6
0 2 3
1 2 6
0 4 6
1 1 6
0 1 6
1 4 6
2 2
1 3
0 2 2
1 1 2
Output:
5 3 5 11
1

-----Explanation-----
Example case 1.The states of array A after each Update-operation:
A: = [2 1 4 10 3 44]
A: = [2 1 4 5 1 22]
A: = [1 1 2 1 1 11]
user: Today is rose day, batch mates of Kabir and Tara decided to celebrate this day by exchanging roses with each other. 
Note:$Note:$ exchanging means both the boy and the girl will give rose to each other. 
In the class there are B$B$ boys and G$G$ girls. 
Exchange of rose will take place if and only if at least one of them hasn't received a rose from anyone else and a rose can be exchanged only once.
Tara has to bring maximum sufficient roses for everyone and is confused as she don't know how many roses to buy.You are a friend of Kabir, so help him to solve the problem so that he can impress Tara by helping her. 

-----Input:-----
- First line will contain T$T$, number of test cases. 
- Each test case contains two space separated integers B$B$ (Number of boys) and  G$G$ (Number of Girls).

-----Output:-----
For each test case, output in a single line the total number of roses exchanged.

-----Constraints:-----
- 1≤T≤105$1 \leq T \leq 10^5$
- 1≤B≤109$1 \leq B \leq 10^{9}$
- 1≤G≤109$1 \leq G \leq 10^{9}$

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

-----Sample Output:-----
8
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-----
- $1 \leq T \leq 10$
- $1 \leq K \leq 10$

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

-----Sample Output:-----
0 
0 
1 1 
0 
1 1 
2 3 5 
0 
1 1 
2 3 5 
8 13 21 34  

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

Suppose there is a circle. There are N Juice shops on that circle. Juice shops are numbered 0 to N-1 (both inclusive). You have two pieces of information corresponding to each of the juice shop:
(1) the amount of Juice that a particular Juice shop can provide and 
(2) the distance from that juice shop to the next juice shop.

Initially, there is a man with a bottle of infinite capacity carrying no juice. He can start the tour at any of the juice shops. Calculate the first point from where the man will be able to complete the circle. Consider that the man will stop at every Juice Shop. The man will move one kilometer for each litre of the juice.

-----Input-----
- 
The first line will contain the value of N.
				
- 
The next N lines will contain a pair of integers each, i.e. the amount of juice that a juice shop can provide(in litres) and the distance between that juice shop and the next juice shop.
				

-----Output-----

An integer which will be the smallest index of the juice shop from which he can start the tour.
		

-----Constraints-----
- 
					1 ≤ N ≤ 105
- 
					1 ≤ amt of juice, distance  ≤ 109

-----Sample Input-----
3

1 5

10 3

3 4

-----Sample Output-----
1

-----Explanation-----

			He can start the tour from the SECOND Juice shop.
		 p { text-align:justify }
user: All strings in Chefland are beautiful because they are binary strings (a binary string contains only characters '0' and '1'). The beauty of a binary string $S$ is defined as the number of pairs $(i, j)$ ($1 \le i \le j \le |S|$) such that the substring $S_i, S_{i+1}, \ldots, S_j$ is special.
For a binary string $U$, let's denote the number of occurrences of the characters '1' and '0' in $U$ by $cnt_1$ and $cnt_0$ respectively; then, $U$ is special if $cnt_0 = cnt_1 \cdot cnt_1$.
Today, Chef's friend Araspa is celebrating her birthday. Chef wants to give Araspa the most beautiful binary string he can find. Currently, he is checking out binary strings in a shop, but he needs your help to calculate their beauties. Tell Chef the beauty of each binary string he gives you.

-----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 string $S$.

-----Output-----
For each test case, print a single line containing one integer — the beauty of the string $S$.

-----Constraints-----
- $1 \le T \le 10$
- $1 \le |S| \le 10^5$
- each character of $S$ is '0' or '1'

-----Example Input-----
2
010001
10

-----Example Output-----
4
1

-----Explanation-----
Example case 1: The special substrings correspond to $(i, j) = (1, 2), (1, 6), (2, 3), (5, 6)$.
user: Do you know that The Chef has a special interest in palindromes? Yes he does! Almost all of the dishes in his restaurant is named by a palindrome strings. The problem is that a name of a dish should not be too long, so The Chef has only limited choices when naming a new dish.

For the given positive integer N, your task is to calculate the number of palindrome strings of length not exceeding N, that contain only lowercase letters of English alphabet (letters from 'a' to 'z', inclusive). Recall that a palindrome is a string that reads the same left to right as right to left (as in "radar").

For example:

- For N = 1, we have 26 different palindromes of length not exceeding N:
"a", "b", ..., "z".
- For N = 2 we have 52 different palindromes of length not exceeding N:
"a", "b", ..., "z",
"aa", "bb", ..., "zz".
- For N = 3 we have 728 different palindromes of length not exceeding N:
"a", "b", ..., "z",
"aa", "bb", ..., "zz",
"aaa", "aba", ..., "aza",
"bab", "bbb", ..., "bzb",
...,
"zaz", "zbz", ..., "zzz".

Since the answer can be quite large you should output it modulo 1000000007 (109 + 7). Yes, we know, most of you already hate this modulo, but there is nothing we can do with it :)

-----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 only line of each test case contains a single integer N.

-----Output-----

For each test case, output a single line containing the answer for the corresponding test case.

-----Constrains-----

- 1 ≤ T ≤ 1000
- 1 ≤ N ≤ 109

-----Example-----
Input:
5
1
2
3
4
100

Output:
26
52
728
1404
508533804

-----Explanation-----

The first three examples are explained in the problem statement above.
user: In this problem, you will be given a polynomial, you have to print what it becomes after differentiation.

Following are the rules for differentiation:
- For a polynomial f(x), its differentiation is defined as f'(x).
- If a is a constant, then differentiation of af(x) is af'(x).
- If f(x) = h(x) + g(x) , then f'(x) = h'(x) + g'(x) 
- If f(x) = x n, then f'(x) = nxn-1. This is true for all  n ≠ 0 .
- If f(x) = c, where c is a constant, f'(x) = 0.

If you are still uncomfortable with differentiation, please read the following:
-  Link to Wikihow page
- Link to Wikipedia entry.

-----Input-----

First line contains T, the number of test cases to follow. 

Each test case contains the follows, the first line contains N, the number of non zero terms in the polynomial. Then N lines follow, each line contains a pair of integer which denotes a term in the polynomial, where the first element denotes the coefficient (a) and the second denotes the exponent (p) of the term.

-----Output-----
Print the polynomial after differentiation in the desired format as described below.
-  If the coefficient of a term in the output polynomial is 3, and the corresponding exponent is 2, print it as 3x^2
- Print " + " (with single space on both side) between each output term.
-  Print the terms in decreasing value of exponent.
-  For the constant term (if any), you have to just print the coefficient. You should not print x^0.

-----Constraints-----
- 1 ≤ T ≤ 10
-   Subtask 1 (20 points)

- 1 ≤ N ≤ 3
- 1 ≤ a ≤ 10
- 0 ≤ p ≤ 10
- Subtask 2 (80 points)

- 1 ≤ N ≤ 10000
- 1 ≤ a ≤ 100000000
- 0 ≤ p ≤ 100000000
-  No two inputs in a test case will have the same exponent.

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

Output:
2x^1
3x^2 + 1
user: Petya is preparing a problem for a local contest in his school. The problem is to find a longest increasing subsequence in a given permutation. A permutation of size n$n$ is a sequence of n$n$ numbers a1,…,an$a_1, \ldots, a_n$ such that every number from 1$1$ to n$n$ occurs in the sequence exactly once. An increasing subsequence of length k$k$ of the sequence a1,…,an$a_1, \ldots, a_n$ is a sequence of indices i1,…,ik$i_1, \ldots, i_k$ such that 1≤i1<…<ik≤n$1 \leq i_1 < \ldots < i_k \leq n$ and ai1<…<aik$a_{i_1} < \ldots < a_{i_k}$. A longest increasing subsequence is an increasing subsequences with the largest length. Note that in general there may be several longest increasing subsequences.
Petya had some tests prepared, but then lost the input part for some of them. He now has a test for a certain value of n$n$, and a sequence i1,…,ik$i_1, \ldots, i_k$ that is supposed to be a longest increasing subsequence. Petya now has to reconstruct a permutation of size n$n$ with this sequence being an answer. Petya doesn't want to take any risks, so he additionally wants this sequence to be the only longest increasing subsequence, that is, all other increasing subsequences have to have strictly smaller length. Help Petya determine if this is possible, and if so, construct any such permutation.

-----Input:-----
The first line contains an integer T$T$, denoting number of test cases.
The first line of every test case contains two integers n$n$ and k−$k-$  the size of the permutation and the length of the longest increasing subsequence (1≤k≤n≤105$1 \leq k \leq n \leq 10^5$).
The second line contains k$k$ integers i1,…,ik−$i_1, \ldots, i_k-$ the longest increasing subsequence (1≤i1<…<ik≤n$1 \leq i_1 < \ldots < i_k \leq n$).

-----Output:-----
If there is no permutation with the sequence i1,…,ik$i_1, \ldots, i_k$ being the only longest increasing subsequence, print NO . Otherwise, print YES on the first line, followed by n$n$ numbers describing any suitable permutation on the second line.

-----Constraints-----
- 1≤T≤10$1 \leq T \leq 10$
- 1≤k≤n≤105$1 \leq k \leq n \leq 10^5$ 
- 1≤i1<…<ik≤n$1 \leq i_1 < \ldots < i_k \leq n$ 

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

-----Sample Output:-----
YES
2 3 1
NO
user: Today, Chef decided to cook some delicious meals from the ingredients in his kitchen. There are $N$ ingredients, represented by strings $S_1, S_2, \ldots, S_N$. Chef took all the ingredients, put them into a cauldron and mixed them up.
In the cauldron, the letters of the strings representing the ingredients completely mixed, so each letter appears in the cauldron as many times as it appeared in all the strings in total; now, any number of times, Chef can take one letter out of the cauldron (if this letter appears in the cauldron multiple times, it can be taken out that many times) and use it in a meal. A complete meal is the string "codechef". Help Chef find the maximum number of complete meals he can make!

-----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 $i$ ($1 \le i \le N$), the $i$-th of these lines contains a single string $S_i$.

-----Output-----
For each test case, print a single line containing one integer — the maximum number of complete meals Chef can create.

-----Constraints-----
- $1 \le T \le 100$
- $1 \le N \le 100$
- $|S_1| + |S_2| + \ldots + |S_N| \le 1,000$
- each string contains only lowercase English letters

-----Example Input-----
3
6
cplusplus
oscar
deck
fee
hat
near
5
code
hacker
chef
chaby
dumbofe
5
codechef
chefcode
fehcedoc
cceeohfd
codechef

-----Example Output-----
1
2
5

-----Explanation-----
Example case 1: After mixing, the cauldron contains the letter 'c' 3 times, the letter 'e' 4 times, and each of the letters 'o', 'd', 'h' and 'f' once. Clearly, this is only enough for one "codechef" meal.
Example case 2: After mixing, the cauldron contains the letter 'c' 4 times, 'o' 2 times, 'd' 2 times, 'e' 4 times, 'h' 3 times and 'f' 2 times, which is enough to make 2 meals.