meetforever/GRPO_CL huggingface dataset:
user: It's year 2018 and it's Christmas time! Before going for vacations, students of Hogwarts School of Witchcraft and Wizardry had their end semester exams.
$N$ students attended the semester exam. Once the exam was over, their results were displayed as either "Pass" or "Fail" behind their magic jacket which they wore. A student cannot see his/her result but can see everyone else's results. Each of $N$ students count the number of passed students they can see.
Given the number of "Pass" verdicts that each of the $N$ students counted, we have to figure out conclusively, the number of students who failed, or report that there is some inconsistency or that we cannot be sure.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- The first line of each test case will contain $N$, representing the number of students who attended the exam.
- Next line contains  $N$ spaced integers representing the number of "Pass" counted by each of the $N$ students.

-----Output:-----
- For each test case, output the answer in a single line. 
- If the counts reported by the students are not consistent with each other or if it's not possible to predict the number of failed students from the given input, then print -1.

-----Constraints-----
- $1 \leq T \leq 50$
- $1 \leq N \leq 10^{5}$
- $0 \leq$ Count given by each Student $\leq 10^{5}$

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

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

-----EXPLANATION:-----
There are 4 students, and they counted the number of passed students as 3,2,2,2. The first student can see that all others have passed, and all other students can see only 2 students who have passed. Hence, the first student must have failed, and others have all passed. Hence, the answer is 1.
user: Write a program that reads two numbers $X$ and $K$. The program first finds the factors of $X$ and then gives the sum of $K$th power of every factor. The program also finds the factor of $k$ and outputs the sum of $X$ times of every factor. 

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase contains of a single line of input, two integers $X, R$. 

-----Output:-----
For each testcase, output in a single line the factors of $X$ and the $K$th power of every factor, seperated by a space.

-----Constraints-----
- $1 \leq T \leq 1000$
- $2 \leq X, K \leq 10^9$

-----Sample Input:-----
1
8 6

-----Sample Output:-----
266304 88

-----EXPLANATION:-----
Factors of x = 8 are 2, 4, and 8. Also, factors of k = 6 are 2, 3, and 6.
2^6 + 4^6 + 8^6 = 266304 and 2 × 8 + 3 × 8 + 6 × 8 = 88.
(Where a ^b denotes a raised to the power of b).
user: Chef solved so many hard questions, now he wants to solve some easy problems for refreshment. Chef asks Cheffina for the new question. Cheffina challenges the chef to print the total number of 0's in the binary representation of N(natural number).

-----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, $N$. 

-----Output:-----
For each test case, output in a single line answer.

-----Constraints-----
- $1 \leq T \leq 10^6$
- $1 \leq N \leq 10^6$

-----Sample Input:-----
2
2
4

-----Sample Output:-----
1
2

-----EXPLANATION:-----
For 1) Binary representation of 2 is 10. i.e. only one 0 present in it.
For 2) Binary representation of 4 is 100, i.e. two 0's present in it.
user: Chef is playing a game with his childhood friend. He gave his friend a list of N numbers named $a_1, a_2 ....  a_N$ (Note: All numbers are unique). Adjust the numbers in the following order:
$(i)$ swap every alternate number with it's succeeding number (If N is odd, do not swap the last number i.e. $a_N$ ).
$(ii)$ add %3 of every number to itself.
$(iii)$ swap the ith number and the (N-i-1) th number.
After this, Chef will give a number to his friend and he has to give the nearest greater and smaller number to it.
If there is no greater or lesser number, put -1.
Help his friend to find the two numbers.

-----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, an integers $N$. 
- Next line contains $N$ integers separated by a space. 
- Next line contains a number to be found, $M$. 

-----Output:-----
For each test case, output in a single line answer given the immediate smaller and greater number separated by a space.

-----Constraints-----
- $1 \leq T \leq 1000$
- $3 \leq N \leq 10^5$
- $1 \leq N_i \leq 10^9$
- $1 \leq M \leq 10^9$

-----Sample Input:-----
1
10
5 15 1 66 55 32 40 22 34 11
38

-----Sample Output:-----
35 41

-----Explaination:-----
Step 1: 15 5 66 1 32 55 22 40 11 34
Step 2: 15 7 66 2 34 56 23 41 13 35
Step 3: 35  13 41 23 56 34 2 66 7 15
35 is the number lesser than 38 and 41 is the number greater than 38 in the given set of numbers.
user: Arya and Aryan live in a country called Nadaca. Nadaca consists of $N$ cities numbered $1$ through $N$, which are connected by some bidirectional roads. Each road has a positive (not necessarily integer) length. Arya lives in city $1$ and Aryan lives in city $N$.
Arya claims that for each city $v$, the shortest path from city $1$ to city $v$ has length $a_v$. Aryan does not disagree, but claims that for each city $v$, the shortest path from city $N$ to city $v$ has length $b_v$. You are the wisest person they know, so they asked you to tell them if it is possible for their claims to be true, i.e. if a road network which fully satisfies their claims exists. Help them!

-----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 $a_1, a_2, \ldots, a_N$.
- The third line contains $N$ space-separated integers $b_1, b_2, \ldots, b_N$.

-----Output-----
For each test case, print a single line containing the string "Yes" if Arya's and Aryan's claims can be true or "No" otherwise.

-----Constraints-----
- $1 \le T \le 10^3$
- $2 \le N \le 10^6$
- $0 \le a_i, b_i \le 10^9$ for each valid $i$
- the sum of $N$ over all test cases does not exceed $10^6$

-----Subtasks-----
Subtask #1 (50 points): the sum of $N$ over all test cases does not exceed $10^3$
Subtask #2 (50 points): original constraints

-----Example Input-----
2
3
0 1 1
1 1 0
3
0 1 5
5 1 0

-----Example Output-----
Yes
No
user: Cherry has a string S$S$ consisting of lowercase English letters. Using this string, he formed a pyramid of infinite length with certain rules:
- N$N$-th row of pyramid contains N$N$ characters.
- Each row of pyramid begins with the first character of the string.
- The subsequent characters of the row are appended to the string in cyclic fashion, until the size of string for that Row is reached (See example pyramid for better understanding).
He has another string T$T$ of smaller (or equal) size.
You are asked Q$Q$ queries. Each query is provided with a row number N$N$. The answer to the query is number of occurrences of string T$T$ in that particular row of pyramid. No of occurrences of String T$T$ in a string V$V$ would mean that you'd need to find number of substrings Vi,Vi+1...Vj$V_i, V_{i+1} ... V_j$ which are equal to String T$T$, where i≤j$i \leq j$.
For eg: If the string is code, then the pyramid will be of the form:
c
co
cod
code
codec
codeco
codecod
codecode
codecodec
codecodeco
...

-----Input:-----
- The first line contains string S$S$ — consisting of lowercase English letters.
- The second line contains string T$T$ — consisting of lowercase English letters.
- Next line contains an integer Q$Q$ — the number of queries.
- Then follow Q$Q$ lines with queries descriptions. Each of them contains a single integer N$N$ denoting the row number of pyramid.

-----Output:-----
- Print Q$Q$ lines. The i$i$-th of them should contain a integer denoting occurrences of string T$T$ in that particular row.

-----Constraints-----
- 1≤|S|≤105$1 \leq |S| \leq 10^5$
- 1≤|T|≤|S|$1 \leq |T| \leq |S|$
- 1≤Q≤105$1 \leq Q \leq 10^5$
- 1≤N≤109$1 \leq N \leq 10^9$

-----Sample Input:-----
codechef
chefcode
3
4
12
1455

-----Sample Output:-----
0
1
181

-----Explanation:-----
Pyramid will be formed as explained in the statement.
Query 1: Row number 4 of the pyramid is code. The number of occurrences of chefcode in code is 0.
Query 2: Row number 12 of the pyramid is codechefcode. The number of occurrences of chefcode in codechefcode is 1.
user: Chef is a brilliant university student that does not attend lectures because he believes that they are boring and coding is life! However, his university follows certain rules and regulations, and a student may only take an exam for a course if he has attended at least 75% of lectures for this course.
Since you are Chef's best friend, you want to help him reach the attendance he needs to take exams. Unfortunately, Chef is still focused on his code and refuses to attend more lectures, so the only option is to have some of his friends mark him as present by proxy. This trick is well-known in the university, but only few have the talent to pull it off.
In a certain course, there is exactly one lesson per day over the course of $D$ days (numbered $1$ through $D$). You are given a string $S$ with length $D$ describing the lessons Chef attended — for each valid $i$, the $i$-th character of this string is either 'A' if Chef was absent on day $i$ or 'P' if Chef was actually present on day $i$.
For each day $d$ when Chef is absent, one of Chef's friends can mark him as present by proxy on this day only if he was present (if he was really present, not just marked as present) on at least one of the previous two days, i.e. days $d-1$ and $d-2$, and on at least one of the following two days, i.e. days $d+1$ and $d+2$. However, it is impossible to mark him as present by proxy on the first two days and the last two days.
Find the minimum number of times Chef has to be marked as present by proxy so that his attendance becomes at least 75% ($0.75$). Chef's attendance is number of days when he was marked as present, either by proxy or by actually being present, divided by $D$.

-----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 $D$.
- The second line contains a single string $S$ with length $D$.

-----Output-----
For each test case, print a single line containing one integer — the minimum number of times Chef needs to be marked as present by proxy, or $-1$ if it is impossible to make Chef achieve 75% attendance.

-----Constraints-----
- $1 \le T \le 200$
- $1 \le D \le 1,000$
- $S$ contains only characters 'A' and 'P'

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

-----Example Input-----
1
9
PAAPPAPPP

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

-----Explanation-----
Example case 1: With a proxy on the third day, the attendance string is "PAPPPAPPP". Now, Chef's attendance is at least 75%, so the minimum number of times Chef needs to be marked as present by proxy is $1$.
user: Snakeland is a well organised city. The houses of the city are organised in an orderly rectangular fashion with dimensions 2 * n, i.e. there are total two rows and n columns. The house in the i-th row and j-th column is also said to be the house at coordinates (i, j). Some of the houses are occupied by snakes, while the others are empty. You are given this information through an array s of dimension 2 * n, where, if s[i][j] = '*', it denotes that there is a snake in the house at coordinates (i, j), and if s[i][j] = '.', it denotes that the house is empty. 
These snakes are planning a coup against a mongoose who controls their city from outside. So, they are trying their best to meet with other snakes and spread information about the date of the coup. For spreading this information, they can just hiss from their house and usually their hiss is so loud that it will be heard in all the cells except if there is a soundproof fence built that cuts the voice. Note that the external borders of Snakeland are already built of soundproof material. The mongoose got to know about the plan, and he wants to construct sound proof fences along the borders of the houses so that no two people should be able to communicate with each other. The fence can be either vertical or horizontal.  Each fence can be of any length, but the mongoose wants to minimize the number of fences to be built. Find out the minimum number of fences that the mongoose should build.

-----Input-----
The first line of the input contains an integer T denoting number of test cases. The descriptions of the T test cases follow.
The first line of each test case contains a single integer, n.
Each of the next two lines contains n characters denoting the first and the second rows of Snakeland respectively.

-----Output-----
For each test case, output a single integer corresponding to the minimum number of fences that the mongoose needs to build.

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ n ≤ 105

-----Example-----
Input
3
2
**
**
3
***
*..
3
*..
.*.

Output
2
3
1

-----Explanation-----
All the examples are shown in the pictures. The fences built are shown by red colored horizontal or vertical segments. You can see that after putting these borders no snake can talk with any another snake.
user: Andrii is good in Math, but not in Programming. He is asking you to solve following problem: Given an integer number N and two sets of integer A and B. Let set A contain all numbers from 1 to N and set B contain all numbers from N + 1 to 2N. Multiset C contains all sums a + b such that a belongs to A and b belongs to B. Note that multiset may contain several elements with the same values. For example, if N equals to three, then A = {1, 2, 3}, B = {4, 5, 6} and C = {5, 6, 6, 7, 7, 7, 8, 8, 9}. Andrii has M queries about multiset C. Every query is defined by a single integer q. Andrii wants to know the number of times q is contained in C. For example, number 6 is contained two times, 1 is not contained in C at all. 

Please, help Andrii to answer all the queries.

-----Input-----

The first line of the input contains two integers N and M. Each of the next M line contains one integer q, the query asked by Andrii.

-----Output-----
Output the answer for each query in separate lines as in example.

-----Constraints-----
- 1 ≤ N ≤ 109
- 1 ≤ M ≤ 105
- 1 ≤ q ≤ 3N

-----Example-----
Input:
3 5
6
2
9
7
5
Output:
2
0
1
3
1
user: There are $N$ cells numbered 1,2,….., N, and every cell has some positive value. you will be given an array $A_1$,$A_2$,…,$A_N$ where $A_i$ is the value of $ith$ cell and an integer $K$.
There is a monkey who wants to reach the $right$ side of the last ($Nth$) cell, initially, the monkey was left side of the first ($1st$) cell. 
In the first step, he can jump into any cell numbered 1,2,….., K.Once he jumped into any cell $i$ next he can only jump to the cell $j$ such that-
- j>i
- j-i <= K  
- A[i]%2 == A[j]%2
You have to find out what is the minimum number of steps required to reach the right side of the cell or In case it's not possible to reach the right side of the cell your answer should be $-1$.NOTE: As soon as monkey reach any cell whose distance from the last cell is less than K, then in the last step, he will jump out to the right side of the ($Nth$) cell.

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- The first line of each test case contains two space-separated integers $N$ and $K$.
- The second line contains $N$ space-separated integers $A_1$,$A_2$,…,$A_N$

-----Output:-----
For each test case, output in a single line minimum steps or $-1$.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq N \leq 2*10^5$
- $1 \leq K \leq N$
- $0 \leq A_i \leq 10^6$
- $\text{the sum of N over all test cases does not exceed $5*10^{5}$}$

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

-----Sample Output:-----
2 

-----EXPLANATION:-----
in the first step, you can jump into any cell 1 or 2 and then in the second step you will jump 
out to the right side of 2nd cell
remember we don't want to reach $Nth$ cell we want to reach the right side of the nth cell.
user: Given an array of size N$N$ and two integers K$K$ and S$S$, the special sum of a subarray is defined as follows:

(Sum of all elements of the subarray) * (K$K$ - p$p$ * S$S$)

Where p$p$ = number of distinct prime factors of “product of all elements of the subarray”.

Find the maximum special sum by considering all non-empty subarrays of the given array.

-----Input-----
- First line contains 3 integers N$N$, K$K$ and S$S$.     
- Second line contains N$N$ integers, the elements of the array.  

-----Output-----
Output a single integer. The maximum special sum considering all non-empty subarrays of the array.

-----Constraints:-----
- 1≤N,K,S≤105$ 1 \leq N, K, S \leq 10^5 $ 
- 0≤K/S≤20$ 0 \leq K / S \leq 20 $ 
- 1<$ 1 < $ Any element of array <105$ < 10^5 $

-----Sample Input-----
4 10 2

14 2 7 15

-----Sample Output-----
138

-----Sample Explanation-----
Consider the subarray {14, 2, 7}

Total number of distinct prime factors in it is 2 (2 and 7).

Therefore, value of special sum is (14 + 2 + 7) * (10 - 2 * 2) = 138.

This is the subarray with the maximum special sum.
user: Zonal Computing Olympiad 2012, 26 Nov 2011

It's dinner time in Castle Camelot, and the fearsome Knights of the Round Table are clamouring for dessert. You, the chef, are in a soup.  There are N knights, including King Arthur, each with a different preference for dessert, but you cannot afford to make desserts for all of them.

You are given the cost of manufacturing each Knight's preferred dessert–since it is a round table, the list starts with the cost of King Arthur's dessert, and goes counter-clockwise.

You decide to pick the cheapest desserts to make, such that for every pair of adjacent Knights, at least one gets his dessert. This will ensure that the Knights do not protest.<p>A strange feature of the Knights is that they will not complain about not getting dessert unless they get support from both their neighbours.  So, you decide to pick the cheapest desserts to make, such that for every pair of adjacent Knights, at least one gets his dessert.</p>

What is the minimum cost of tonight's dinner, given this condition?

For instance, suppose there are 5 Knights and their desserts cost 1, 2, 1, 2 and 2.  In this case, the minimum cost is 4, which you can achieve by feeding the first, third and fourth (or fifth) Knights.  

-----Input format-----
There are 2 lines of input.  The first line contains a single integer N, the number of seats at the table. The next line contains N space separated integers, each being the cost of the dessert of a Knight, listed in counterclockwise order around the table, starting with King Arthur.

-----Output format-----
The output should be a single line containing a single integer, the minimum possible cost for you, the chef.

-----Testdata-----
Each Knight's dessert costs strictly more than 0 and strictly less than 1000.  You may assume that 1 ≤ N ≤ 106.  In 30% of the test cases, 1 ≤ N ≤ 103.

- Subtask 1 (30 marks)
- Subtask 2 (70 marks)

-----Sample Input-----
5
1 2 1 2 2

-----Sample Output-----
4
user: -----Problem Statement-----
Chef has a sequence of N segments: [L1, R1], [L2, R2], ..., [LN, RN]. He wants to transform the first segment to the last one (with index N). His plan is to do this big deal with a number of transformations: firstly he will transform

the first segment to the second one, then to the third one, then to the fourth one, and so on till N-th one.

Chef can use operation of a single type: shift one segment border by one unit. So, if he has segment [L, R], he can transform it into one of the following segments: [L + 1, R] (we will denote such operation with string L+), [L, R + 1] (will be denoted as R+), [L - 1, R] (L-), [L, R - 1] (R-). Chef doesn't like empty segments, therefore he cannot use any operation that makes a segment empty (L = R).

Chef really wants to transform his segment as fast as possible. Please, help him. Find the sequence with minimal number of operations that transforms his segment. If there are multiple such sequences pick the lexicographically minimal one.

-----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 Chef has. 

The following N lines contain pairs of integers. The i-th line contains integers Li, Ri, denoting i-th Chef's segment.

-----Output-----
For each test case print an answer - two lines. The first line should contain the minimal number of operations. The second line should contain the sequence of operations

without any whitespaces.

-----Constraints-----
- 1 ≤ T, N ≤ 1000.
- -1000 ≤ Li < Ri ≤ 1000.
The total sum of N values for all test cases doesn't exceed 1000.

-----Example-----
Input:
4
3
-1 0
0 1
3 5
1
0 1
3
-2 -1
-2 -1
-2 0
4
4 6
3 5
-1 1
1 2

Output:
9
R+L+R+L+R+L+R+L+R+
0

1
R+
13
L-R-L-L-L-L-R-R-R-R-L+R+L+
user: Chef had an array A with length N, but some of its elements got lost. Now, each element of this array is either unknown (denoted by -1) or a positive integer not exceeding K.
Chef decided to restore the array A by replacing each unknown element by a positive integer not exceeding K.
However, Chef has M restrictions that must hold for the restored array. There are two types of restrictions:

- I L R, meaning that for each i such that L < i ≤ R, the condition Ai - Ai-1 = 1 should be satisfied.
- D L R, meaning that for each i such that L < i  ≤ R, the condition Ai - Ai-1 = -1 should be satisfied.

Chef would like to know the number of ways to restore the array while satisfying all restrictions, modulo 109+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 line of each test case contains three space-separated integers N, M and K.
- The second line contains N integers A1, A2, ..., AN.
- Each of the following M lines contains one restriction in the form I L R or D L R.

-----Output-----
For each test case, print a single line containing one integer - the number of ways to restore the array modulo 109+7.

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ N, M ≤ 100,000
- 1 ≤ K ≤ 1,000,000,000
- 1 ≤ L < R ≤ N
- 1 ≤ Ai ≤ K or Ai = -1 for each valid i
- 1 ≤ sum of N over all test cases ≤ 500,000
- 1 ≤ sum of M over all test cases ≤ 500,000

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

3
4 2 10
2 3 5 4
I 1 2 
D 3 4
5 2 10
-1 -1 -1 -1 -1
I 1 3
D 3 5
6 2 2
-1 -1 -1 -1 -1 -1
I 1 4
D 4 6

Output:

1
8
0
user: Note : This question carries $150$ $points$
There is an outcry in Middle Earth, as the greatest war between Orgs of Dark Lord Sauron and Frodo Baggins is about to begin. To end the war, Frodo decides to destroy the ring in the volcano of Mordor. There are many ships that lead Frodo to Mordor, and he is confused about which one he should board.  Given two-ship numbers $M$ and $N$, Frodo has to solve a problem to find the ship which he should board.
Find the number of pairs (x, y), such that $1<=x<=M$ and $1<=y<=N$, for which $x*y + x+ y = string(x)+string(y)$ is true. 
Also, calculate the number of distinct x satisfying the given condition.  The number of pairs and the number of distinct x will help select Frodo the boat he should board. Help Frodo defeat Sauron. 

-----Input :-----
- First line contains $T$ as number of test cases 
- Each test case contains two integers $M$ and $N$ 

-----Output :-----
- For each test case, print two integers - the number of such pairs (x,y) and the number of distinct x

-----Constraints :-----
- 1 ≤ T ≤ 5000
- 1 ≤  M, N ≤ 10^9

-----Sample Input :-----
1
1 9

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

-----Explanation :-----
For test case two M=1 and N=9  Pair (1,9) satisfies the above condition 1*9+1+9= “19” and only x=1 satisfies the equation.