meetforever/GRPO_CL huggingface dataset:
user: Cyael is a teacher at a very famous school in Byteland and she is known by her students for being very polite to them and also to encourage them to get good marks on their tests.
Then, if they get good marks she will reward them with candies :) However, she knows they are all very good at Mathematics, so she decided to split the candies evenly to all the students she considers worth of receiving them, so they don't fight with each other. 
She has a bag which initially contains N candies and she intends to split the candies evenly to K students. To do this she will proceed as follows: while she has more than K candies she will give exactly 1 candy to each student until she has less than K candies. On this situation, as she can't split candies equally among all students she will keep the remaining candies to herself.
Your job is to tell how many candies will each student and the teacher
receive after the splitting is performed.

-----Input-----
The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.
Each test case will consist of 2 space separated integers, N and K denoting the number of candies and the number of students as described above.

-----Output-----
For each test case, output a single line containing two space separated integers, the first one being the number of candies each student will get, followed by the number of candies the teacher will get.

-----Constraints-----
-  T<=100 in each test file 
- 0 <= N,K <= 233 - 1 

-----Example-----Input:
2
10 2
100 3

Output:
5 0
33 1

-----Explanation-----
For the first test case, all students can get an equal number of candies and teacher receives no candies at all 
For the second test case, teacher can give 33 candies to each student and keep 1 candy to herselfUpdate:
There may be multiple whitespaces before, after or between the numbers in input.
user: There are N hills in a row numbered 1 through N from left to right. Each hill has a height; for each valid i, the height of the i-th hill is Hi. Chef is initially on the leftmost hill (hill number 1). He can make an arbitrary number of jumps (including zero) as long as the following conditions are satisfied:

- Chef can only jump from each hill to the next hill, i.e. from the i-th hill, he can jump to the i+1-th hill (if it exists).
- It's always possible to jump to a hill with the same height as the current hill.
- It's possible to jump to a taller hill if it's higher than the current hill by no more than U.
- It's possible to jump to a lower hill if it's lower than the current hill by no more than D.
- Chef can use a parachute and jump to a lower hill regardless of its height (as long as it's lower than the current hill). This jump can only be performed at most once.

Chef would like to move as far right as possible. Determine the index of the rightmost hill Chef can reach.

-----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, U and D.
- The second line contains N space-separated integers H1, H2, ..., HN.

-----Output-----
For each test case, print a single line containing one integer — the index of the rightmost reachable hill.

-----Constraints-----
- 1 ≤ T ≤ 100
- 1 ≤ N ≤ 100
- 1 ≤ U, D ≤ 1,000,000
- 1 ≤ Hi ≤ 1,000,000 for each valid i

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

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

3
5 3 2
2 5 2 6 3
5 2 3
4 4 4 4 4
5 2 7
1 4 3 2 1

Output:

3
5
1

-----Explanation-----
Example case 1: Chef can jump to second hill because it's higher by no more than U=3 than first hill, to jump to third hill Chef has to use parachute 
because it's lower than second hill by 3 which is more than D=2, Chef can't jump to fourth hill because it's higher than third hill by 4 which is more than U=3
Example case 2: All hills are of the same height, so chef can reach the last hill with no problems.
Example case 3: Chef can't jump to second hill because it's too high for him
user: Since due to COVID 19, India has undergone a complete 21 day lockdown. So Amol was attending an online lecture where his professor asked him to solve a question. Amol was unable to solve the question so he asked you to solve the question and give him the correct answer. 

The question was asked a such that his professor gave him a number M and a list of integers of length N (i.e. A1, A2,..... AN) and you have to find out all the subsets that add up to M and the total number of subsets will be the final answer. 

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 of each test case contains a single integer M. 

• The third line contains N space-separated integers A1, A2,..... AN. 

Output: 
For each test case, print a single line containing one integer ― the no. of subsets that adds upto M.

Constraints: 
• 1≤T≤5

• 1≤N≤100 

• 10≤M≤100 

• 1≤Ai≤100

ExampleInput:

2

4

16

2 4 6 10 

4 

20 

2 8 12 10 

Output:

2

2

Explanation:
In the 1st example there are two subsets {2,4,10} and {6,10} that adds upto 16 so the output is 2.
user: You are given a rooted tree on N vertices. The nodes are numbered from 1 to N, and Node 1 is the root. Each node u has an associated value attached to it: Au.
For each vertex v, we consider the path going upwards from v to the root. Suppose that path is v1, v2, .., vk, where v1 = v and vk = 1. The cost of any node on this path is equal to the minimum value among all the nodes to its left in the path sequence, including itself. That is, cost(vi) = min1 <= j <= i{Avj}. And the cost of the path is the sum of costs of all the nodes in it.
For every node in the tree, find the cost of the path from that node to the root.

-----Input-----
- The first line of the input contains a single integer, N, denoting the number of nodes in the tree.
- The next line contains N-1 integers, the i-th of which denotes the parent of node i+1.
- The next line contains N integers, the i-th of which denotes Ai.

-----Output-----
Output a single line containing N integers, the i-th of which should be the cost of the path from node i to the root.

-----Constraints-----
- 1 ≤ N ≤ 100,000
- -1,000,000,000 ≤ Av ≤ 1,000,000,000

-----Subtasks-----
- Subtask #1 (30 points): 1 ≤ N ≤ 2000
- Subtask #2 (70 points): Original constraints.

-----Example-----
Input:
8
1 1 1 1 5 8 6
1 2 3 4 5 15 70 10

Output: 
1 3 4 5 6 21 96 26

-----Explanation-----
For example, take a look at the path from vertex 7: The path is 7 -> 8 -> 6 -> 5 -> 1.
Cost(7) has no choice but to be A7. So Cost(7) = 70.
Cost(8) will be minimum of A7 and A8, which turns out to be A8. So Cost(8) = 10.
Cost(6) = minimum {A7, A8, A6} = minimum {70, 10, 15} = 10.
Cost(5) = minimum {70, 10, 15, 5} = 5.
Cost(1) = minimum {70, 10, 15, 5, 1} = 1. 
So, the cost of the path from 7 to the root is 70 + 10 + 10 + 5 + 1 = 96.
user: You may have helped Chef and prevented Doof from destroying the even numbers. But, it has only angered Dr Doof even further. However, for his next plan, he needs some time. Therefore, Doof has built $N$ walls to prevent Chef from interrupting him. You have to help Chef by telling him the number of walls he needs to destroy in order to reach Dr Doof.
Formally, the whole area can be represented as the first quadrant with the origin at the bottom-left corner. Dr. Doof is located at the origin $(0, 0)$. There are $N$ walls, the i-th wall is a straight line segment joining the points $(a_i, 0)$ and $(0, a_i)$. For every initial position of Chef $(x_j, y_j)$, find the number of walls he needs to break before reaching Doof. Obviously, chef can't start from a point on the wall. Therefore, if $(x_j, y_j)$ lies on any of the given walls, print $-1$ in a new line.

-----Input-----
- First line contains $T$, denoting the number of testcases.
- The first line of every test case contains a single integer $N$ denoting the number of walls Dr Doof has built.
- The next line contains $N$ space separated distinct integers each denoting $a_i$.
- The next line contains a single integer $Q$ denoting the number of times Chef asks for your help.
- The next $Q$ lines contains two space separated integers $x_j$ and $y_j$, each denoting the co-ordinates of the starting point of Chef.

-----Output-----
For each query, print the number of walls Chef needs to break in order to reach Dr Doof in a separate line. If Chef tries to start from a point on any of the walls, print $-1$.

-----Constraints-----
- $1 \leq T \leq 2 * 10^2$
- $1 \leq N, Q \leq 2 * 10^5$
- $1 \leq a_i \leq 10^9$
- $0 \leq x_j, y_j \leq 10^9$
- $a_1 < a_2 < a_3 < .... < a_N$
- Sum of $N$ and $Q$ over all testcases for a particular test file does not exceed $2 * 10^5$

-----Sample Input-----
1
2
1 3
5
0 0
2 0
0 4
1 1
1 2

-----Sample Output-----
0
1
2
1
-1

-----Explanation-----
The sample input can be represented by the graph given below:

If Chef starts from $(0, 0)$, he can reach Dr Doof without destroying any wall.

If Chef starts from $(2, 0)$, he has to destroy the $1st$ wall.

If Chef starts from $(0, 4)$, he has to destroy both the walls.

If Chef starts from $(1, 1)$, he has to destroy the $1st$ wall.

As $(1, 2)$ lies on the second wall, the answer is $-1$ for the last query.
user: You are given a sequence $A_1, A_2, \ldots, A_N$. You want all the elements of the sequence to be equal. In order to achieve that, you may perform zero or more moves. In each move, you must choose an index $i$ ($1 \le i \le N$), then choose $j = i-1$ or $j = i+1$ (it is not allowed to choose $j = 0$ or $j = N+1$) and change the value of $A_i$ to $A_j$ — in other words, you should replace the value of one element of the sequence by one of its adjacent elements.
What is the minimum number of moves you need to make in order to make all the elements of the sequence equal?

-----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 minimum required number of moves.

-----Constraints-----
- $1 \le T \le 100$
- $1 \le N \le 100$
- $1 \le A_i \le 100$ for each valid $i$

-----Example Input-----
3
5
1 1 1 1 1
4
9 8 1 8
2
1 9

-----Example Output-----
0
2
1

-----Explanation-----
Example case 1: No moves are needed since all the elements are already equal.
Example case 3: We can perform one move on either $A_1$ or $A_2$.
user: -----
HALLOWEEN EVE
-----

In some other world, today is Halloween Eve.There are N trees planted in Mr. Smith’s
garden. The height of the i-th tree (1≤i≤N) is h i meters.
He decides to choose K trees from these trees and decorate them with electric lights.
To make the scenery more beautiful, the heights of the decorated trees should be as
close to each other as possible.

	

More specifically, let the height of the tallest decorated tree be hmax meters, and the
height of the shortest decorated tree be hmin meters.
The smaller the value hmax−hmin is, the better. What is the minimum possible value of
hmax−hmin?

		
		

-----Constraints-----

	    2≤K< N ≤105 

1≤hi≤109

hi is an integer
	

-----Input Format-----

Input is given from Standard Input in the following format:

N K 

h1 

h2 

: 

hN

-----Output-----

Print the minimum possible value of hmax−hmin.

		
		

-----Example Text Case-----
Input:

5 3
10
15
11
14
12

Output:

2

Explanation

If we decorate the first, third and fifth trees, hmax=12,hmin=10 so hmax−hmin=2. This is
optimal.
user: You are given an $array$ of size $N$ and an integer $K$ ( $N > 1 , K > 0$ ).
Each element in the array can be incremented by $K$ or decremented by $K$ $at$ $most$ $once$.
So there will be  $3^n$ possible combinations of final array. (As there are 3 options for every element).

Out of these combinations, you have to select a combination, in which the $absolute$ difference between the largest and the smallest element is $maximum$.
You have to print the $maximum$ $absolute$ $difference$.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase contains of a two lines of input
- First line contains two integers $N, K$. 
- Second line contains $N$ space separated integers.

-----Output:-----
For each testcase, output the maximum absolute difference that can be achieved on a new line.

-----Constraints-----
- $1 \leq T \leq 10$
- $2 \leq N \leq 1000$
- $1 \leq K , arr[i]  \leq 10000$
$NOTE$:  Large  input files, Use of fastio is recommended.

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

4 3

4 2 5 1

3 5

2 5 3

-----Sample Output:-----
10

13
user: Mr. Pr and Ms. Ad are at $a$ and $b$ respectively on an infinite number line. Mr. Pr wants to meet Ms. Ad.
Mr. Pr can choose to move  $c$ or $d$ units in 1 second. If Mr. Pr moves $c$ units then Ms. Ad will move $d$ units and vice versa. (Both of them always moved in positive x-direction)
You have to determine if Mr. Pr can meet with Ms. Ad after some integral amount of time, given that Mr. Pr chooses optimally. Note that meeting after a fractional amount of time does not count.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase contains four space separated integers, $a$, $b$, $c$, and $d$.

-----Output:-----
- For each test case, output a single line containing "YES" if Mr. Pr meets with Ms. Ad, otherwise "NO".

-----Constraints-----
- $1 \leq T \leq 10^5$
- $1 \leq a,b,c,d \leq 10^9$

-----Sample Input:-----
2
3 4 1 2
10 20 3 7

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

-----Explanation:-----
In the first test case, Mr. Pr will move 2 units in the first second and Ms. Ad moves 1 unit simultaneously and they meet. 
In the second test case, it is impossible to meet (fractional time is not allowed).
user: Zonal Computing Olympiad 2012, 26 Nov 2011

We consider sequences of opening and closing brackets with two types of brackets, () and [].  A bracket sequence is well-bracketed if we can pair up each opening bracket with a matching closing bracket in the usual sense.  For instance, the sequences (), [] ([]) and []([]) are well-bracketed, while (, ()], (], )( and [(]) are not well-bracketed.  In the last case, each opening bracket has a matching closing bracket and vice versa, but the intervals spanned by the different types of brackets intersect each other instead of being contained one within the other.

The alternating depth of a well-bracketed sequence tells us the maximum number of times we switch between the two types of brackets when we have inner matched brackets enclosed within outer matched brackets.  For instance, the alternating depth of (), [[[]]] and ()[][] is 1, the alternating depth of [()] and ()([]) is 2, the alternating depth of ([()]) and [()][(([]))] is 3, and so on.

Given a well-bracketed sequence, we are interested in computing three quantities.
- The alternating depth of the sequence.
- The maximum number of symbols between any pair of matched brackets of the type ( and ), including both the outer brackets.
- The maximum number of symbols between any pair of matched brackets of the type [ and ], including both the outer brackets.

For instance, the alternating depth of (([]))[[[()]]] is 2, the maximum number of symbols between a matched pair () is 6 and the maximum number of symbols between a matched pair [] is 8.



-----Input format-----
The input consists of two lines.  The first line is a single integer N, the length of the bracket sequence. Positions in the sequence are numbered 1,2,…,N. The second line is a sequence of N space-separated integers that encode the bracket expression as follows: 1 denotes an opening bracket (, 2 denotes a closing bracket ), 3 denotes an opening bracket [ and 4 denotes a closing bracket ].  Nothing other than 1, 2, 3 or 4 appears in the second line of input and the corresponding expression is guaranteed to be well-bracketed.

-----Output format-----
Your program should print 3 space-separated integers in a line, denoting the three quantities asked for in the following order: alternating depth, length of the maximum sequence between matching () brackets and length of the maximum sequence between matching [] brackets.  

-----Testdata-----
You may assume that 2 ≤ N ≤ 105.  In 30% of the test cases, 2 ≤ N ≤ 103.

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

-----Sample Input-----
14
1 1 3 4 2 2 3 3 3 1 2 4 4 4

-----Sample Output-----
2 6 8
user: Chef recently graduated Computer Science in university, so he was looking for a job. He applied for several job offers, but he eventually settled for a software engineering job at ShareChat. Chef was very enthusiastic about his new job and the first mission assigned to him was to implement a message encoding feature to ensure the chat is private and secure.
Chef has a message, which is a string $S$ with length $N$ containing only lowercase English letters. It should be encoded in two steps as follows: 
- Swap the first and second character of the string $S$, then swap the 3rd and 4th character, then the 5th and 6th character and so on. If the length of $S$ is odd, the last character should not be swapped with any other.
- Replace each occurrence of the letter 'a' in the message obtained after the first step by the letter 'z', each occurrence of 'b' by 'y', each occurrence of 'c' by 'x', etc, and each occurrence of 'z' in the message obtained after the first step by 'a'.
The string produced in the second step is the encoded message. Help Chef and find this message.

-----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 the message string $S$.

-----Output-----
For each test case, print a single line containing one string — the encoded message.

-----Constraints-----
- $1 \le T \le 1,000$
- $1 \le N \le 100$
- $|S| = N$
- $S$ contains only lowercase English letters

-----Example Input-----
2
9
sharechat
4
chef

-----Example Output-----
shizxvzsg
sxuv

-----Explanation-----
Example case 1: The original message is "sharechat". In the first step, we swap four pairs of letters (note that the last letter is not swapped), so it becomes "hsraceaht". In the second step, we replace the first letter ('h') by 's', the second letter ('s') by 'h', and so on, so the resulting encoded message is "shizxvzsg".
user: Chuck Norris just created a universe. He has already started the space-time continuum. Now he needs to fill the universe with matter and antimatter for the avatar cycle to start.
He created $N$ distinct pairs of elementary matter and their corresponding antimatter particles. Annihilations happen only when a matter particle is placed adjacent to its corresponding antimatter particle and not when it is placed adjacent to any other matter or antimatter particle. Thus for every distinct particle of matter in the universe there exist one and only one antimatter particle that can cause its annihilation.
Chuck Norris wants to arrange all the particles in a 2-D circular pattern such that a matter particle is placed adjacent only to antimatter particles and vice versa. In how many distinct ways can he arrange them, regardless of the orientation of the arrangement, such that total annihilation of ALL particles doesn't take place i.e., $at least$ $one$ pair remain unannihilated after he arranges them in a circular pattern.
Since the total number of distinct ways can be very large, print the result modulo $1000000007$.

-----Input:-----
- First line has single integer $T$, i.e., the number of test cases
- The first and only line of each test case has a single integer $N$ i.e., number of distinct pairs of matter-antimatter particles.

-----Output:-----
- For of each test case print a single line, i.e., total number of distinct ways.

-----Constraints :-----
- $1 \leq T \leq 1000$
- $0 \leq N \leq 10^7$

-----Subtasks-----
- 
Subtask 1 : 10 pts
- $1 \leq T \leq 100$
- $0 \leq N \leq 100$ 
- 
Subtask 2 : 90 pts
- Original Constraints

-----Sample input :-----
1
3

-----Sample output :-----
8

-----Explanation:-----
For 3 pairs of elementary matter and their corresponding  antimatter particles, Chuck Norris can arrange them in 8 ways such that at least one pair of matter antimatter particles survive annihilation.
user: You are given a grid of size M x N, where each square is colored with some random color among K colors with each having equal probability.

A Good Rectangle is defined as one where all squares lying on the inner border are of the same color.

What is the expected number of Good Rectangles in the given grid.

-----Input-----

- 
First Line contains M, N, K

-----Output-----
A single value rounded off to the nearest Integer corresponding to the required answer.

-----Constraints-----
-  1 <= N <= 105 
-  1 <= M <= 105 
-  1 <= K <= 105 

-----Example-----
Input:
1 3 1
Output:
6
user: Jem is famous for his laziness at school. He always leaves things to last minute. Now Jem has N problems in the assignment of "Advanced topics in algorithm" class to solved. The assignment is due tomorrow and as you may guess he hasn't touch any of the problems. Fortunately he got a plan as always.

The first step will be buying a pack of Red Bull and then to work as hard as he can. Here is how he is going to spend the remaining time:

Jem will not take a break until he finishes at least half of the remaining problems. Formally, if N is even then he will take he first break after finishing N / 2 problems. If N is odd then the break will be after he done (N + 1) / 2 problems. Each of his break will last for B minutes. Initially, he takes M minutes in solving a problem, after each break he will take twice more time in solving a problem, i.e. 2 * M minutes per problem after the first break.

Jem will start working soon and ask you to help him calculate how much time it will take until he finish the last problem!

-----Input-----

The first line contains a single integer T represents the number of test cases in the input.
Each line in the next T line contains three integers N, B and M represents a test case.

-----Output-----

For each test case output a single line containing an integer represent how much time Jem will need (in minutes).

-----Constraints-----
- 1 ≤ T ≤ 100
- 1 ≤ N, B, M ≤ 108

-----Example-----
Input:
2
9 1 2
123456 123456 123456

Output:
45
131351258112

-----Explanation-----

In the first test case, Jem will proceed as below:

- Initially, Jem has 9 problems to solve. since it is an odd number, Jem will finish the first (9 + 1) / 2 = 5 problems with speed of 2 minutes/problem.
- After that, Jem takes 1 minute break.
- Now he has 4 problems to solve, which is an even number, so Jem will solve the next 4 / 2 = 2 problems. his speed after the first break has now became 4 minutes/problem.

- Again, he takes a 1 minute break.
- he has now 2 problems left so he do one more problem in 8 minutes.
- He takes 1 minute break.
- he solves the last problem in 16 minutes.

So, Jem will need time = 5 × 2 + 1 + 2 × 4 + 1 + 8 + 1 + 16  = 45