meetforever/GRPO_CL huggingface dataset:
user: The median of a sequence is the element in the middle of the sequence after it is sorted. For a sequence with even size, the median is the average of the two middle elements of the sequence after sorting. For example, for a sequence $A = [1, 3, 3, 5, 4, 7, 11]$, the median is equal to $4$, and for $A = [2, 3, 4, 5]$, the median is equal to $(3+4)/2 = 3.5$.
Fulu is a famous programmer in his country. He wrote the following program (given in pseudocode) for finding the median in a sequence $A$ with length $N$:
read(N)
read(A[0], A[1], ..., A[N-1])
sort(A)
# simultaneously remove elements A[K] and A[N mod K] from the array A
# the order of the remaining N-2 elements is unchanged
remove(A[K], A[N mod K])
return A[(N-2)/2] # integer division

The program takes an integer $K$ as a parameter. Fulu can choose this integer arbitrarily between $1$ and $N-1$ inclusive.
Little Lima, Fulu's friend, thinks Fulu's program is wrong (as always). As finding one counterexample would be an easy task for Lima (he has already found the sequence $A = [34, 23, 35, 514]$, which is a counterexample for any $K \le 3$), Lima decided to make hacking more interesting. Fulu should give him four parameters $S, K, m, M$ (in addition to $N$) and Lima should find the lexicographically smallest proper sequence $A$ with length $N$ as a counterexample.
We say that a sequence $A$ with length $N$ ($0$-indexed) is proper if it satisfies the following conditions:
- it contains only positive integers
- $A_0 + A_1 +A_2 + \dots + A_{N-1} = S$
- $m \le A_i \le M$ for each $0 \le i < N$
- the number returned by Fulu's program, ran with the given parameter $K$, is different from the correct median of the sequence $A$
Can you help Lima find the lexicographically smallest counterexample or determine that Fulu's program works perfectly for the given parameters?

-----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 five space-separated integers $N$, $S$, $K$, $m$ and $M$.

-----Output-----
For each test case, if there is no proper sequence, print a single line containing the integer $-1$; otherwise, print a single line containing $N$ space-separated integers — the lexicographically smallest proper sequence.

-----Constraints-----
- $1 \le T \le 8$
- $3 \le N \le 10^5$
- $1 \le K \le N-1$
- $1 \le S \le 10^9$
- $1 \le m \le M \le S$

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

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

-----Explanation-----
Example case 1: For the sequence $A = [1, 1, 4]$, it is already sorted, so the program would just remove the first two elements ($K=1$, $N\%K=0$) and return the only remaining element $4$. Obviously, the median of the original sequence is $1$. It can be shown that there is no lexicographically smaller proper sequence.
Example case 2: The only possible sequence is $A = [1, 1, 1, 1]$, which is not proper, since Fulu's program will give the correct answer $1$ for it.
user: Shashank is playing a game with his friends.
There are n sticks located in a row at points $a_1,a_2, ...,a_n$. Each stick has a height- $h_i$. A person can chop a stick down, after which it takes over one of the regions [$a_i$ - $h_i$, $a_i$] or [$a_i$, $a_i$ + $h_i$]. The stick that is not chopped remains at the point $a_i$. A person can chop a stick in a particular direction if the region to be taken up by the chopped stick does not overlap with an already existing point. The winner $($s$)$ of the game will be one or more people who can answer the question: What is the maximum number of sticks that can be chopped?
Shashank wants to win the game and hence he needs needs your help in finding out what is the maximum number of sticks that can be chopped down.

-----Input:-----
- The first line of each input contains a single integer n.
- n lines follow. Each of the n lines contain a pair of integers: $a_i,h_i$.

-----Output:-----
Output in a single line answer- the maximum number of sticks that can be chopped down.

-----Constraints-----
- $1 \leq n \leq 10^5$
- $1 \leq a_i,h_i \leq 10^9$
- The pairs are given in the order of ascending $a_i$. No two sticks are located at the same point.

-----Sample Input 1:-----
5
1 2
2 1
5 10
10 9
19 1

-----Sample Input 2:-----
5
1 2
2 1
5 10
10 9
20 1

-----Sample Output 1:-----
3

-----Sample Output 2:-----
4

-----EXPLANATION:-----
In the first example you can fell the sticks as follows:
- chop the stick 1 to the left — now it will take over the region $[ - 1;1]$
- chop the stick 2 to the right — now it will take over the region $[2;3]$
- spare the stick 3— it will take over the point $5$
- spare the stick 4— it will take over the point $10$
- chop the stick 5 to the right — now it will take over the region $[19;20]$
user: To help Lavanya learn all about binary numbers and binary sequences, her father has bought her a collection of square tiles, each of which has either a 0 or a 1 written on it. Her brother Nikhil has played a rather nasty prank. He has glued together pairs of tiles with 0 written on them. Lavanya now has square tiles with 1 on them and rectangular tiles with two 0's on them, made up of two square tiles with 0 stuck together). Thus, she can no longer make all possible binary sequences using these tiles.
To amuse herself, Lavanya has decided to pick a number $N$ and try and construct as many binary sequences of length $N$ as possible using her collection of tiles. For example if $N$ = 1, she can only make the sequence 1. For $N$=2, she can make 11 and 00. For $N$=4, there are 5 possibilities: 0011, 0000, 1001, 1100 and 1111.
Lavanya would like you to write a program to compute the number of arrangements possible with $N$ tiles so that she can verify that she has generated all of them. Since she cannot count beyond 15746, it is sufficient to report this number modulo 15746.

-----Input:-----
A single line with a single integer $N$.

-----Output:-----
A single integer indicating the number of binary sequences of length $N$, modulo 15746, that Lavanya can make using her tiles.

-----Constraints:-----
You may assume that $N \leq$ 1000000.

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

-----Sample Output:-----
5

-----Explanation:-----
This corresponds to the example discussed above.
user: The chef is playing a game of long distance. Chef has a number K and he wants to find the longest distance between the index of the first and the last occurrence of K in a given array of N numbers.

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains two lines of input.
- Next line with Two integers in one line $K, N$.
- Next line with $N$ space-separated integers.

-----Output:-----
For each test case, output in a single line answer as the index of first and last occurrence of K in the given array.
Note: Here Indexing is from 1 not  0 based.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq k \leq 10^5$
- $1 \leq N \leq 10^5$

-----Sample Input:-----
2
2 6
2 3 4 2 1 6
4 6
2 3 4 2 1 6

-----Sample Output:-----
3
0

-----EXPLANATION:-----
For 1) Index of First and last occurrence of 2 in the given array is at 1 and 4, i.e. distance is 3. 
For 2) 4 occurs only once in the given array hence print 0.
user: Vasya's older brother, Petya, attends an algorithm course in his school. Today he learned about matchings in graphs. Formally, a set of edges in a graph is called a matching if no pair of distinct edges in the set shares a common endpoint.
Petya instantly came up with an inverse concept, an antimatching. In an antimatching, any pair of distinct edges should have a common endpoint.
Petya knows that finding a largest matching in a graph is a somewhat formidable task. He wonders if finding the largest antimatching is any easier. Help him find the number of edges in a largest antimatching in a given graph.

-----Input:-----
The first line contains T$T$, number of test cases per file.
The first line of each test case contains two integers n$n$ and m−$m-$ the number of vertices and edges of the graph respectively (1≤n≤104$1 \leq n \leq 10^4$, 0≤m≤104$0 \leq m \leq 10^4$).
The next m$m$ lines describe the edges. The i$i$-th of these lines contains two integers ui$u_i$ and vi−$v_i-$ the indices of endpoints of the i$i$-th edge (1≤ui,vi≤n$1 \leq u_i, v_i \leq n$, ui≠vi$u_i \neq v_i$).
It is guaranteed that the graph does not contain self-loops nor multiple edges. It is not guaranteed that the graph is connected.

-----Output:-----
Print a single number per test case −$-$ the maximum size of an antichain in the graph.

-----Constraints-----
- 1≤T≤10$1 \leq T \leq 10$
- 1≤n≤104$1 \leq n \leq 10^4$
- 0≤m≤104$0 \leq m \leq 10^4$
- 1≤ui,vi≤n$1 \leq u_i, v_i \leq n$
- ui≠vi$u_i \neq v_i$

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

-----Sample Output:-----
3
1
0

-----EXPLANATION:-----
In the first sample all three edges form an antimatching.
In the second sample at most one of the two edges can be included in an antimatching since they do not share common endpoints.
In the third sample there are no edges, hence the answer is 0$0$.
user: Chef got in the trouble! He is the king of Chefland and Chessland. There is one queen in Chefland and one queen in Chessland and they both want a relationship with him. Chef is standing before a difficult choice…
Chessland may be considered a chessboard with $N$ rows (numbered $1$ through $N$) and $M$ columns (numbered $1$ through $M$). Let's denote a unit square in row $r$ and column $c$ by $(r, c)$. Chef lives at square $(X, Y)$ of this chessboard.
Currently, both queens are living in Chessland too. Each queen, when alone on the chessboard, can see all squares that lie on the same row, column or diagonal as itself. A queen from $(x_q, y_q)$ cannot see a square $(r, c)$ if the square $(X, Y)$ is strictly between them. Of course, if the queens can see each other, the kingdom will soon be in chaos!
Help Chef calculate the number of possible configurations of the queens such that the kingdom will not be in chaos. A configuration is an unordered pair of distinct squares $(x_{q1}, y_{q1})$ and $(x_{q2}, y_{q2})$ such that neither of them is the square $(X, Y)$. Two configurations are different if the position of queen $1$ is different or the position of queen $2$ is different.

-----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 four space-separated integers $N$, $M$, $X$ and $Y$. 

-----Output-----
For each test case, print a single line containing one integer — the number of configurations such that the kingdom will not be in chaos.

-----Constraints-----
- $1 \le T \le 1000$
- $1 \le X \le N \le 10^2$
- $1 \le Y \le M \le 10^2$
- $2 \le N, M$

-----Example Input-----
2
3 3 2 2
4 4 2 3

-----Example Output-----
24
94

-----Explanation-----
Example case 1: Half of these configurations are:
- $(1, 1), (3, 3)$
- $(1, 1), (2, 3)$
- $(1, 1), (3, 2)$
- $(1, 2), (3, 3)$
- $(1, 2), (3, 2)$
- $(1, 2), (3, 1)$
- $(1, 3), (3, 1)$
- $(1, 3), (3, 2)$
- $(1, 3), (2, 1)$
- $(2, 1), (2, 3)$
- $(2, 1), (1, 3)$
- $(2, 1), (3, 3)$
user: 2021 was approaching and the world was about to end. So 2 gods Saurabhx and Saurabhy (from Celesta) created the Cyberverse. But this time disappointed with humans both the gods decided not to have humans in this world. So they created a world of cyborgs. A world without humans. Isn’t it interesting? So let us dive into the cyberverse and have a look at their problems.
There are $N$ kid cyborgs with Chief Cyborg '100gods' and he has $K$ weapons with him. He wants to distribute those $K$ weapons among $N$ kid cyborgs. Since all the kid cyborgs are very good friends, so they set a rule among themselves for taking those weapons. The rule states that the difference between kid cyborg having the maximum weapons and the kid cyborg having minimum weapons should be less than or equal to $1$.

Find the value of the minimum number of weapons a kid cyborg can have when all the $K$ weapons are distributed among them.

-----Input:-----
- The first line contains an integer $T$, denoting the number of test cases.
- Each of the next $T$ lines will contain two space-separated integers denoting $N$ and $K$ respectively.

-----Output:-----
- For each test case ,output a single line containing an integer $X$ denoting the minimum number of weapons a kid cyborg can have in that test case.  

-----Constraints:-----
- $1 \leq T \leq 10^5$
- $1 \leq N \leq 10^5$
- $1 \leq K \leq 10^9$

-----Sample Input:-----
1
5 8

-----Expected Output:-----
1

-----Explanation-----
- There are $5$ kids and $8$ weapons.  
- Hence we will distribute the weapons such that $3$ kids have $2$ weapons each and the remaining $2$ kids have $1$ weapon each.  
- Hence the minimum number of weapons a kid cyborg has is $1$. (  That is, $min(1,2)$ = $1$ )
user: Zonal Computing Olympiad 2015, 29 Nov 2014

We say that two integers x and y have a variation of at least K, if |x − y| ≥ K (the absolute value of their difference is at least K). Given a sequence of N integers a1,a2,...,aN and K, the total variation count is the number of pairs of elements in the sequence with variation at least K, i.e. it is the size of the set of pairs

{(i,j)|1≤i<j≤N and|ai−aj|≥K} 

For example if K = 1 and the sequence is 3,2,4 the answer is 3. If K = 1 and the sequence is 3, 1, 3 then the answer is 2.

Your task is to write a program that takes a sequence and the value K as input and computes the total variation count.

-----Input format-----
The first line contains two positive integers N and K, separated by a space.

This is followed by a line containing N integers separated by space giving the values
of the sequence.

-----Output format-----
A single integer in a single line giving the total variation count.

-----Test data-----
You may assume that all integers in the input are in the range 0 to 10^8 inclusive.

Subtask 1 (40 marks) : 1 ≤ N ≤ 4000, 1 ≤ K ≤ 10^8 
Subtask 2 (60 marks) : 1 ≤ N ≤ 65000, 1 ≤ K ≤ 10^8 

-----Sample Input-----
3 1 
3 1 3

-----Sample Output-----
2
user: Bob has got some injury in his leg and due to this he can take exactly M steps in one move. 
Bob enters a square field of size NxN. The field is only having one gate(for both entrance and exit) at its one of the corners. Bob started walking along the perimeter of square field.(remember Bob can only take exactly M steps in one move and cannot reverse his direction of motion).
Bob wants to know how many minimum number of moves he needs to come out(i.e. he reaches the same gate from where he entered into the field) from the square field. Tell the answer to Bob ASAP.
Luckily, you came to know M=N+1.

-----Input-----
- The first line of the input contains an integer T denoting the number of test cases.
-  Each test case contains a single integer N denoting the sides of the square.

-----Output-----
- For each test case, output a single line containing minimum number of moves Bob required to come out from the field.

-----Constraints-----
- 1 ≤ T ≤ 10000
- 1 ≤ N ≤ 1000000000

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

Output:
2
8

-----Explanation-----
Example case 1.Let four corners of square be (0,0), (0,1), (1,1), (1,0). Let gate be at (0,0). Bob takes 2 steps in one move. Let movement of Bob be as follows (0,0) -> (1,1) -> (0,0). Thus minimum moves needed were 2. 
Example case 2.Let four corners of square be (0,0), (0,2), (2,2), (2,0). Let gate be at (0,0). Bob takes 3 steps in one move. Let movement of Bob be as follows (0,0) -> (2,1) -> (0,2) -> (1,0) -> (2,2) -> (0,1) -> (2,0) -> (1,2) -> (0,0). Thus minimum number of moves needed are 8.
user: Chef has a sequence of positive integers $A_1, A_2, \ldots, A_N$. He wants to choose some elements of this sequence (possibly none or all of them) and compute their MEX, i.e. the smallest positive integer which does not occur among the chosen elements. For example, the MEX of $[1, 2, 4]$ is $3$.
Help Chef find the largest number of elements of the sequence $A$ which he can choose such that their MEX is equal to $M$, or determine that it is impossible.

-----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 $M$.
- The second line contains $N$ space-separated integers $A_1, A_2, \ldots, A_N$.

-----Output-----
For each test case, print a single line containing one integer ― the maximum number of elements Chef can choose, or $-1$ if he cannot choose elements in such a way that their MEX is $M$.

-----Constraints-----
- $1 \le T \le 100$
- $2 \le M \le N \le 10^5$
- $1 \le A_i \le 10^9$ for each valid $i$
- the sum of $N$ over all test cases does not exceed $10^6$

-----Example Input-----
1
3 3
1 2 4

-----Example Output-----
3

-----Explanation-----
Example case 1: The MEX of whole array is 3. Hence, we can choose all the elements.
user: "Humankind cannot gain anything without first giving something in return. To obtain, something of equal value must be lost. That is alchemy's first law of Equivalent Exchange. In those days, we really believed that to be the world's one, and only truth."
-- Alphonse Elric
Now, here we have an equivalent exchange law for triangles which states that two right-angled isosceles triangles of the same color can be made into a square of the same color using Alchemy.
You are given N$N$ right-angled isosceles colored triangles numbered from 1$1$ to N$N$. For each triangle, the two equal sides have a length of 1$1$ unit. The Color of i$i$-th triangle is given by Ci$C_i$.
To create a tower, we choose some consecutive (2×k)+1$2 \times k)+1$ triangles for any k≥0$k \geq 0$. We then pick some 2×k$2 \times k$ of them (these need not be consecutive), and form k$k$ pairs of triangles such that both triangles in pair have the same color. Also, each of the 2×k$2 \times k$ should be in exactly one pair. Then the two triangles in each pair are joined using Alchemy (following the law of equivalent exchange for triangles) to form squares and these k$k$ squares are placed one upon other. The one remaining triangle is placed as a roof to the tower. This results in a tower of the height of k$k$.
Find the maximum height of the tower that can be formed. 
In other words, you should select the largest consecutive segment of triangles, such that you can form a tower using every single one of those triangles. In particular, you leave out one triangle, which will form the roof, and the other triangles should all be paired up such that both triangles in a pair have the same colour.

-----Input:-----
- The first line contains T$T$, the number of test cases. Then the test cases follow. 
- For every test case, the first line contains N$N$ denoting the number of triangles.
- For every test case, the second line contains N$N$ space-separated integers Ci$C_{i}$ denoting the color of the triangles. ( 1≤i≤N$1 \leq i \leq N$).

-----Output:-----
For every test case, output a single integer denoting the maximum height of the tower that can be formed.

-----Constraints-----
- 1≤T≤100$1 \leq T \leq 100$ 
- 1≤N≤105$1 \leq N \leq 10^{5}$ 
- 1≤Ci≤30$1 \leq C_{i} \leq 30$ 
- Sum of N$N$ over all test cases doesn't exceed 5×105$5\times 10^{5}$ 

-----Sample Input:-----
4
14
5 4 2 2 3 2 1 3 2 7 4 9 9 9
3
1 2 1
3
1 1 1
5
1 2 3 4 1

-----Sample Output:-----
3
1
1
0

-----EXPLANATION:-----
- #1$1$: The subarray [2,2,3,2,1,3,2]$[2, 2, 3, 2, 1, 3, 2]$ results in a tower of height 3$3$.
- #2$2$: The subarray [1,2,1]$[ 1, 2, 1 ]$ results in a tower of height 1$1$.
- #3$3$: The subarray [1,1,1]$[ 1, 1, 1 ]$ results in a tower of height 1$1$. 
- #4$4$: The subarrays [1]$[ 1 ]$, [2]$[ 2 ]$ , [3]$[ 3 ]$, [4]$[ 4 ]$ and [1]$[ 1 ]$ all results in a tower of height 0$0$.

The above tower is possible by subarray [2,2,3,2,1,3,2]$[2, 2, 3, 2, 1, 3, 2]$ resulting in a height of 3$3$ in test case 1$1$.
user: Chef has just finished the construction of his new garden. He has sown the garden with patches of the most beautiful carpet grass he could find. He has filled it with patches of different color and now he wants to evaluate how elegant his garden is.
Chef's garden looks like a rectangular grid of cells with N rows and M columns. So there are N x M cells in total.	In each cell Chef planted grass of some color.
The elegance of the garden is defined by the number of squares, composed of at least four  garden cells, with edges parallel to the sides of the garden, that have four corner cells of the same color.
Given the description of Chef's garden, calculate how many such squares exist.
Input format
The first line contains the number T, the number of test cases. In the following lines,	T test cases follow (without any newlines between them.)
The first line of each test case contains N and M, separated by a single space.
Each of the next N lines contains M characters without any spaces between them, and without any leading or trailing spaces.
Each character describes the color of the corresponding cell in the garden and belongs to the set of lowercase and uppercase lettes of the English alphabet.
One letter in lowercase and uppercase describes different colors.
Output format
For each test case, print the number of squares that conform to the definition in the	problem statement.
Constraints
1 ≤ T ≤ 50

1 ≤ N, M ≤ 50

Sample input
3
2 2
aa
aA
3 3
aba
bab
aba
4 4
aabb
aabb
bbaa
bbaa

Sample output
0
1
4

Explanation
In the first case the only avaliable square does not conform to the definition in the problem statement because 'a' and 'A' describes different colors.
In the second case, you can select the 4 a's at the corners of the garden.
In the third case, you can only make four squares, from the four 2x2 segments	that are of the same color.