meetforever/GRPO_CL huggingface dataset:
user: Chef wants to make a feast. In order to do that, he needs a lot of different ingredients. Each ingredient has a certain tastiness; the tastiness of each ingredient may be any positive integer. Initially, for each tastiness between $K$ and $K+N-1$ (inclusive), Chef has an infinite supply of ingredients with this tastiness.
The ingredients have a special property: any two of them can be mixed to create a new ingredient. If the original ingredients had tastiness $x$ and $y$ (possibly $x = y$), the new ingredient has tastiness $x+y$. The ingredients created this way may be used to mix other ingredients as well. Chef is free to mix ingredients in any way he chooses any number of times.
Let's call a tastiness $v$ ($v > 0$) unreachable if there is no way to obtain an ingredient with tastiness $v$; otherwise, tastiness $v$ is reachable. Chef wants to make ingredients with all reachable values of tastiness and he would like to know the number of unreachable values. Help him solve this problem. Since the answer may be large, compute it modulo $1,000,000,007$ ($10^9+7$).
Note that there is an infinite number of reachable values of tastiness, but it can be proven that the number of unreachable values is always finite for $N \ge 2$.

-----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 two space-separated integers $N$ and $K$.

-----Output-----
For each test case, print a single line containing one integer — the number of unreachable values of tastiness, modulo $1,000,000,007$.

-----Constraints-----
- $1 \le T \le 10^5$
- $2 \le N \le 10^{18}$
- $1 \le K \le 10^{18}$

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

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

-----Example Output-----
0
2

-----Explanation-----
Example case 1: It is possible to obtain ingredients with all values of tastiness.
Example case 2: Ingredients with tastiness $1$ and $2$ cannot be made.
user: Consider a number X on which K Mag-Inc operations are to be performed. In a Mag-Inc operation, the number X undergoes an increment of A/B times of X where A and B are two integers.

There is a numerator and a denominator array of size K which contain the ith values of A and B. After K Mag-Inc operations, the number X turns to M.

Now your job is to find what percentage of M is to be decremented from M if it has to be converted back to X. Let this percentage be denoted by Z.

Print the integral part of Z.

-----Input:-----

First line contains an integer T denoting the number of test cases.

First line of every test case contains two space separated integers X and K.

The second and third line of every test case will contain K space separated integers 

denoting the Numerator and Denominator array.

-----Output:-----

For each test case, print the required result in a single line.

-----Constraints:-----
1 ≤ T ≤ 100
1 ≤ K, A, B ≤ 40000
1≤X≤10^100

-----Example:-----Input:
2
100 1
1 
4
100 2
1 1
2 3Output:
20
50

-----Explanation:-----

Case 2: 100 undergoes an increment of (1/2)*100. Therefore M = 100 + 50.

Now M = 150.

Now again, 

M undergoes an increment of (1/3)*150. Therefore, M = 150 + 50. 

Now as we want to revert back M = 200 to X i.e. 100, we need to decrement it by a value 

of 100 and we know that 100 is 50% of 200.

Hence, we print 50.
user: The chef was busy in solving algebra, he found some interesting results, that there are many numbers which can be formed by sum of some numbers which are prime. Chef wrote those numbers in dairy. Cheffina came and saw what the chef was doing. Cheffina immediately closed chef's dairy and for testing chef's memory, she starts asking numbers and chef needs to answer wheater given number N can be formed by the sum of K prime numbers if it yes then print 1 else print 0. 

-----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, two integers $N, K$.

-----Output:-----
For each test case, output in a single line answer as 1 or 0.

-----Constraints-----
- $1 \leq T \leq 10^5$
- $2 \leq N \leq 10^5$
- $1 \leq K \leq 10^5$

-----Sample Input:-----
2
12 2
11 2

-----Sample Output:-----
1
0
user: In order to win over and get noticed by his favorite streamer Daenerys, Jon decides to donate a significant amount of money . Every donation made to Daenerys is of $at$ $least$ $1$ $beastcoin$ and is displayed on Daenerys's stream alongside any message written and is visible to every viewer.
After spotting that Daenerys had set out a target for the streaming day at minimum $X$   beastcoins, all her viewers would only donate amounts less than $X$ beastcoins. Jon decided to better all of them by straight out donating more than or equal to $X$ beastcoins. Further, he decides to write a message along with his special donation to leave her in awe. His message would be : "Crossing my donation with any other donation will only increase the value of my donation". By Crossing, he means to take the $XOR$ . 
But even for all his intellectual brilliance, money doesn't grow on trees for Jon. After all he is an underpaid employee in his fancy big name MNC. Unlike Daenerys's daily cash cow who makes videos of how she donated carelessly to other people, Jon has a budget and in this case too, he is looking for the minimum donation he needs to make.
Can you tell Jon the minimum amount he needs to donate to Daenerys so that he is able to credibly put out the above comment alongside the donation in order to HOPEFULLY win her over.

-----Input Format-----
- First line contain an interger $T$, which denotes number of testcases. Next $T$ lines contain single interger $X$.    

-----Output Format-----
- For every testcase print one integer, i.e. minimum donation Jon needs to make.

-----Constriants-----
- $ 1 \leq T \leq 100000 $ 
- $ 2 \leq X \leq 10^{18} $ 

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

3

7 

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

8
user: Given an array of n$n$ integers : A1,A2,...,An$ A_1, A_2,... , A_n$, find the longest size subsequence which satisfies the following property: The xor of adjacent integers in the subsequence must be non-decreasing.

-----Input:-----
- First line contains an integer n$n$, denoting the length of the array. 
- Second line will contain n$n$ space separated integers, denoting the elements of the array.

-----Output:-----
Output a single integer denoting the longest size of subsequence with the given property.

-----Constraints-----
- 1≤n≤103$1 \leq n \leq 10^3$
- 0≤Ai≤1018$0 \leq A_i \leq 10^{18}$

-----Sample Input:-----
8
1 200 3 0 400 4 1 7

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

-----EXPLANATION:-----
The subsequence of maximum length is {1, 3, 0, 4, 1, 7} with Xor of adjacent indexes as {2,3,4,5,6} (non-decreasing)
user: Zonal Computing Olympiad 2014, 30 Nov 2013

In IPL 2025, the amount that each player is paid varies from match to match.  The match fee depends on the quality of opposition, the venue etc.

The match fees for each match in the new season have been announced in advance.  Each team has to enforce a mandatory rotation policy so that no player ever plays three matches in a row during the season.

Nikhil is the captain and chooses the team for each match. He wants to allocate a playing schedule for himself to maximize his earnings through match fees during the season.  

-----Input format-----
Line 1: A single integer N, the number of games in the IPL season.
Line 2: N non-negative integers, where the integer in
position i represents the fee for match i.

-----Output format-----
The output consists of a single non-negative integer, the
maximum amount of money that Nikhil can earn during this IPL
season. 

-----Sample Input 1-----
5 
10 3 5 7 3 

-----Sample Output 1-----
23

(Explanation: 10+3+7+3)

-----Sample Input 2-----
8
3 2 3 2 3 5 1 3

-----Sample Output 2-----
17

(Explanation: 3+3+3+5+3)

-----Test data-----
There is only one subtask worth 100 marks.  In all inputs:

• 1 ≤ N ≤ 2×105
• The fee for each match is between 0 and 104, inclusive.

-----Live evaluation data-----
There are 12 test inputs on the server during the exam.
user: The Chef has a huge square napkin of size 2n X 2n. He folds the napkin n-3 times. Each time he folds its bottom side over its top side, and then its right side over its left side. After each fold, the side length of the napkin is reduced by half. The Chef continues folding until there remains a 8x8 sheet, lying flat on a table.
Oh, did I forget to mention that the Chef was cooking a new brown colored curry while folding the napkin. He drops some brown colored gravy onto some cells in the folded 8x8 napkin. When he drops the gravy, it soaks through all the cells below it.
Now the Chef unfolds the napkin to its original size. There are now many curry stained brown colored cells in the napkin. They form several separate regions, each of which is connected. Could you help the Chef count how many regions of brown cells are there in the napkin?
Note that two cells are adjacent if they share a common edge (they are not considered adjacent if they only share a corner). Two cells are connected if we can go from one cell to the other via adjacent cells. A region is a maximal set of cells such that every two of its cells are connected.
Please see the example test case for more details.

-----Input-----
The first line contains t, the number of test cases (about 50). Then t test cases follow. Each test case has the following form:
- The first line contains N (3 ≤ N ≤ 109)
- Then, 8 lines follow. Each line is a string of 8 characters, 0 or 1, where 1 denotes a stained brown cell in the folded napkin.

-----Output-----
For each test case, print a single number that is the number of disconnected brown regions in the unfolded napkin. Since the result may be a very large number, you only need to print its remainder when dividing by 21945.

-----Example-----
Input:
3
3
01000010
11000001
00000000
00011000
00011000
00010100
00001000
00000000
4
01000010
11000001
00000000
00011000
00011000
00010100
00001000
00000000
1000000000
11111111
11111111
11111111
11111111
11111111
11111111
11111111
11111111

Output:
6
22
1

-----Output details-----

Case 1 and 2: There are 6 brown regions in the 8x8 napkin. If we unfold it once, it has 22 brown regions: 11 regions in the top half and 11 regions in the bottom half (as shown in the figure above).
Case 3: All cells of the napkin are stained, so there is always one brown region, no matter how many times the Chef unfolds the napkin.
user: Alice and Bob created $N$ and $M$ recipes, respectively ($N, M \ge 1$), and submitted them to Chef for evaluation. Each recipe is represented by a string containing only lowercase English letters. Let's denote Alice's recipes by $A_1, A_2, \ldots, A_N$ and Bob's recipes by $B_1, B_2, \ldots, B_M$.
Accidentally, Chef mixed up those recipes ― now, he has $L = N+M$ recipes in a sequence $S_1, S_2, \ldots, S_L$. Thankfully, the recipes created by Alice and Bob are distinguishable from each other. It is well-known that for each recipe $s$ created by Alice, the following property holds, and for each recipe created by Bob, it does not hold: For each $1 \le l < r \le |s|$, the substring $s_l, s_{l+1}, \ldots, s_r$ contains at least as many vowels as consonants. The letters 'a', 'e', 'i', 'o', 'u' are vowels, while the other letters are consonants.
The score of a candidate who made $K$ recipes is calculated as the product of $\frac{x_c}{fx_c^K}$ for all letters $c$ that occur in at least one of these recipes; here, $x_c$ is the number of recipes which contain the letter $c$ and $fx_c$ is the total number of occurrences of this letter in all $K$ recipes.
Let's denote the scores of Alice and Bob by $sc_A$ and $sc_B$ respectively. Chef wants to know their ratio $sc_A/sc_B$. We know that Chef is a legendary cook, but he is not very good at calculating, so he is asking you to find that number.

-----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 $L$.
- $L$ lines follow. For each valid $i$, the $i$-th of these lines contains a single string $S_i$.

-----Output-----
For each test case, if the ratio of scores exceeds $10^7$, print a single line containing the string "Infinity" (without quotes); otherwise, print a single line containing one real number $sc_A/sc_B$.
Your answer will be considered correct if its absolute or relative error does not exceed $10^{-6}$. It is guaranteed that $sc_A/sc_B$ does not lie in the range $10^7 \pm 10$.

-----Constraints-----
- $1 \le T \le 10^5$
- $2 \le L \le 10^5$
- $2 \le |S_i| \le 10^5$ for each valid $i$
- for each valid $i$, $S_i$ contains only lowercase English letters
- the sum of $|S_1| + |S_2| + \ldots + |S_L|$ over all test cases does not exceed $10^7$

-----Subtasks-----
Subtask #1 (25 points):
- $L \le 10$
- $|S_i| \le 10$ for each valid $i$
Subtask #2 (75 points): original constraints

-----Example Input-----
2
4
aba
abc
bab
aac
3
aba
baab
abc

-----Example Output-----
1.1250000
0.0277778

-----Explanation-----
Example case 1: The recipes "aba" and "aac" are created by Alice, while the recipes "abc" and "bab" are created by Bob. The scores are:
- $sc_A = \frac{x_a}{fx_a^N} \cdot \frac{x_b}{fx_b^N} \cdot \frac{x_c}{fx_c^N} = \frac{2}{4^2} \cdot \frac{1}{1^2} \cdot \frac{1}{1^2} = \frac{1}{8}$
- $sc_B = \frac{x_a}{fx_a^M} \cdot \frac{x_b}{fx_b^M} \cdot \frac{x_c}{fx_c^M} = \frac{2}{2^2} \cdot \frac{2}{3^2} \cdot \frac{1}{1^2} = \frac{1}{9}$
- $\frac{sc_A}{sc_B} = \frac{1/8}{1/9} = 1.125$
user: Today is Chef's birthday. His mom decided to surprise him with a truly fantastic gift: his favourite binary string B. But, unfortunately, all the stocks of binary string B have been sold out, and only a binary string A (A ≠ B) is available in the market.
She purchases the string A and tries to convert it to string B by applying any of following three operations zero or more times.
AND Operation:
She will choose a pair of indices i and j such that i != j and perform following sequence of operations.

-  result = Ai & Aj 
-  Ai = result & Ai 
-  Aj = result & Aj 

OR Operation:
She will choose a pair of indices i and j such that i != j and perform following sequence of operations.

-  result = Ai | Aj 
-  Ai = result | Ai 
-  Aj = result | Aj 

XOR Operation:
She will choose a pair of indices i and j such that i != j and perform following sequence of operations.

-  result = Ai ^ Aj 
-  Ai = result ^ Ai 
-  Aj = result ^ Aj 

Chef's mom is eagerly waiting to surprise him with his favourite gift and therefore, she wants to convert string A to string B as fast as possible. Can you please help her by telling her the minimum number of operations she will require? If it is impossible to do so, then let Chef's mom know about it.

-----Input-----
First line of input contains a single integer T denoting the number of test cases. T test cases follow.
First line of each test case, will contain binary string A.
Second line of each test case, will contain binary string B.

-----Output-----
For each test case, Print "Lucky Chef" (without quotes) in first line and minimum number of operations required to convert string A to sting B in second line if conversion is possible. Print "Unlucky Chef" (without quotes) in a new line otherwise.

-----Constraints-----
- 1 ≤ T ≤ 105
- 1 ≤ |A| ≤ 106
- 1 ≤ |B| ≤ 106
- A != B
- |A| = |B|
- sum of |A| over all test cases does not exceed 106
- sum of |B| over all test cases does not exceed 106

-----Subtasks-----
-  Subtask #1 (40 points) : Sum of |A| & |B| over all test cases does not exceed 103
-  Subtask #2 (60 points) : Sum of |A| & |B| over all test cases does not exceed 106

-----Example-----
Input
2
101
010
1111
1010
Output
Lucky Chef
2
Unlucky Chef

-----Explanation-----
Example case 1.
- Applying XOR operation with indices i = 1 and j = 2. Resulting string will be 011.
- Then, Applying AND operation with indices i = 1 and j = 3. Resulting string will be 010.

Example case 2.
- It is impossible to convert string A to string B.
user: You are Dastan, the great Prince of Persia!
After searching long for the mysterious 'Sands of Time', you have finally arrived at the gates of the city that hosts the ancient temple of the gods. However, the gate is locked and it can only be opened with a secret code, which you need to obtain by solving the following puzzle:
There is a table in front of you, with $N$ coins placed in a row and numbered $1$ through $N$ from left to right. For each coin, you know whether it is initially showing heads or tails. You have to perform exactly $K$ operations. In one operation, you should remove the rightmost coin present on the table, and if this coin was showing heads right before it was removed, then you should also flip all the remaining coins. (If a coin was showing heads, then after it is flipped, it is showing tails, and vice versa.)
The code needed to enter the temple is the number of coins which, after these $K$ operations are performed, have not been removed and are showing heads. Can you find this number? The fate of Persia lies in your hands…

-----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 two space-separated integers $N$ and $K$.
- The second line contains $N$ space-separated characters. For each valid $i$, the $i$-th of these characters is 'H' if the $i$-th coin is initially showing heads or 'T' if it is showing tails.

-----Output-----
For each test case, print a single line containing one integer ― the number of coins that are showing heads after $K$ operations.

-----Constraints-----
- $1 \le T \le 200$
- $1 \le K < N \le 100$

-----Subtasks-----
Subtask #1 (100 points): original constraints

-----Example Input-----
3
5 3
H T T H T
7 4
H H T T T H H
6 1
T H T H T T

-----Example Output-----
1
2
2
user: During Quarantine Time Chef is at home and he was quite confused about what to cook so, he went to his son and asked about what would he prefer to have? He replied, cakes.
Now, chef cook $N$ number of cake and number of layers for every cake is different. After cakes are baked, Chef arranged them in a particular order and then generates a number by putting number of layers of cakes as digit in sequence (e.g., if chef arranges cakes with layers in sequence $2$, $3$ and $5$ then generated number is $235$).
Chef has to make his son powerful in mathematics, so he called his son and ask him to arrange the cakes in all the possible ways and every time when different sequence is generated he has to note down the number. At the end he has to find sum of all the generated numbers. So, help him to complete this task.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- The first line of each test case contains a single integer $N$ denoting number of cakes.
- The second line contains $N$ space-separated integers $L1$ $L2$ … $LN$ layers of the cake.

-----Output:-----
For each test case, print a single line containing sum of all the possible numbers which is generated by arranging cake in different sequence.

-----Constraints :-----
- $1 \leq T  \leq 2*10^5$
- $1 \leq N, L1, L2, L3,…, LN \leq 9$

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

-----Sample Output:-----
2220

-----Explanation:-----
Sum of all possibilities :
$235 + 532 + 253 + 352 + 523 + 325 = 2220 $
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:-----
5
1
2
3
4
5

-----Sample Output:-----
1
1
32
1
32
654
1
32
654
10987
1
32
654
10987
1514131211

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: Bobby has decided to hunt some Parrots. There are n horizontal branch of trees aligned parallel to each other. Branches are numbered 1 to n from top to bottom. On each branch there are some parrots sitting next to each other. Supposed there are a[i]$a[i]$ parrots sitting on the i−th$ i-th$ branch.  
Sometimes Bobby shots one of the parrot and the parrot dies (suppose that this parrots sat at the i−th$i-th$ branch). Consequently all the parrots on the i−th$i-th$ branch to the left of the dead parrot get scared and jump up on the branch number i − 1$i - 1$, if there exists no upper branch they fly away. Also all the parrots to the right of the dead parrot jump down on branch number i + 1$i + 1$, if there exists no such branch they fly away.  
Bobby has shot m parrots. You're given the initial number of parrots on each branch, tell him how many parrots are sitting on each branch after the shots.  

-----Input:-----
The first line of the input contains an integer N$N$. The next line contains a list of space-separated integers a1, a2, …, an.
The third line contains an integer M$M$. Each of the next M$M$ lines contains two integers x[i]$x[i]$ and y[i]$y[i]$. The integers mean that for the i-th time Bobby shoot the y[i]-th (from left) parrot on the x[i]-th branch. It's guaranteed there will be at least y[i] parrot on the x[i]-th branch at that moment.   

-----Output:-----
On the i−th$i-th$ line of the output print the number of parrots on the i−th$i-th$ branch.

-----Constraints-----
- 1≤N≤100$1 \leq N \leq 100$
- 0≤a[i]≤100$0 \leq a[i] \leq 100$
- 0≤M≤100$0 \leq M \leq 100$
- 1≤x[i]≤n$1 \leq x[i] \leq n$, 1≤y[i]$1 \leq y[i] $

-----Sample Input:-----
5

10 10 10 10 10

5

2 5

3 13

2 12

1 13

4 6   
3

2 4 1

1

2 2   

-----Sample Output:-----
0

12

5

0

16   
3

0

3
user: As you know America’s Presidential Elections are about to take place and the most popular leader of the Republican party Donald Trump is famous for throwing allegations against anyone he meets.

He goes to a rally and meets n people which he wants to offend. For each person i he can choose an integer between 1 to max[i].

He wants to decide in how many ways he can offend all these persons (N) given the condition that all numbers chosen by him for each person are distinct.
So he needs your help to find out the number of ways in which he can do that. If no solution is possible print 0

-----Input-----
The first line of the input contains an integer T (1<=T<=100) 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 people Trump wants to offend. The second line contains N space-separated integers maxnumber[0], maxnumber[1], ..., maxnumber[n-1] denoting the maxnumber that trump can choose for each person. 

-----Output-----
For each test case, output a single line containing the number of ways Trump can assign numbers to the people, modulo 1,000,000,007. If it's impossible to assign distinct integers to the people, print 0

-----Constraints-----
- 1 ≤ T ≤ 100
- 1 ≤ N ≤ 50
- 1 ≤ Maxnumber[i] ≤ 3000

-----Example-----
Input:
3
1
4
2
10 5
4
2 3 1 3

Output:
4
45
0


-----Explanation-----
In case 1, He can choose any number from 1 to 4

In case 2,Out of the total 50 combination he can not take (1,1) ,(2,2) , (3,3) ,(4,4) or (5,5).