meetforever/GRPO_CL huggingface dataset:
user: There are $N$ buckets numbered $1$ through $N$. The buckets contain balls; each ball has a color between $1$ and $K$. Let's denote the number of balls with color $j$ that are initially in bucket $i$ by $a_{i, j}$.
For each $i$ from $1$ to $N-1$ (in this order), someone draws a ball uniformly at random from bucket $i$ and puts it into bucket $i+1$, then continues to draw the next ball. After putting a ball in bucket $N$, this person draws a ball, again uniformly at random, from bucket $N$.
For each color from $1$ to $K$, find the probability that the ball drawn from bucket $N$ has this color.

-----Input-----
- The first line of the input contains two space-separated integers $N$ and $K$. 
- $N$ lines follow. For each $i$ ($1 \le i \le N$), the $i$-th of these lines contains $K$ space-separated integers $a_{i, 1}, a_{i, 2}, \ldots, a_{i, K}$.

-----Output-----
Print a single line containing $K$ space-separated real numbers. For each valid $i$, the $i$-th of these numbers should denote the probability that the last drawn ball has color $i$. your answer will be considered correct if absolute or relative error does not exceed $10^{-6}$

-----Constraints-----
- $1 \le N, K \le 1,000$
- $0 \le a_{i, j} \le 10$ for each valid $i, j$
- initially, there is at least one ball in bucket $1$

-----Subtasks-----
Subtask #1 (30 points): $1 \le N, K \le 100$
Subtask #2 (70 points): original constraints

-----Example Input-----
2 2
0 1
1 1

-----Example Output-----
0.333333 0.666667

-----Explanation-----
user: In this problem we are concerned with words constructed using the lowercase letters of the English alphabet - that is, a,b,c,…,z. These words need not necessarily be meaningful: any sequence of letters forms a word. For example, abbca is a word.
We say that we can "hop" from the word $w_1$ to the word $w_2$ if they are "sufficiently close". We define $w_2$ to be sufficiently close to $w_1$ if one of the following holds:
- $w_2$ is obtained from $w_1$ by deleting one letter.
- $w_2$ is obtained from $w_1$ by replacing one of the letters in $w_1$ by some letter that appears to its right in $w_1$ and which is also to its right in alphabetical order.
For example, we can hop from abbca to abca by deleting the second (or third letter). We can hop from aabca to abbca by replacing the a in the second position by the letter b that appears to the right of the a in aabca and which is also to its right in alphabetical order. On the other hand we cannot hop from abbca to aabca since we would need to replace the b in the second position by a, but a is to the left of b in alphabetical order.
You will be given a collection of words $W$. Your task is to find the length of the longest sequence $w_1, w_2, \ldots $ of distinct words from $W$ such that we may hop from $w_1$ to $w_2$, $w_2$ to $w_3$ and so on. We call this the hopping number for this set.
For example, if
$W$ = {abacd, bcdada, dd, abcd,bcdd, adcd, addd, aa, ccd, add, ad}
then, the hopping number is 7 corresponding to the sequence
abacd, abcd, adcd, addd, add, ad, dd

-----Input Format:-----
- The first line of the input contains one integer $N$ indicating the number of words in the input. 
- This is followed by $N$ lines of input, lines 2, 3,…, $N$+1, each containing one word over the letters {a,b,…, z}.

-----Output Format:-----
The output should be a single integer, indicating the hopping number of the given set of words.

-----Test Data:-----
- $N \leq 100$
- You may assume that each word has at most 10 letters.

-----Sample Input:-----
11
abacd
bcdada
dd
abcd
bcdd
adcd
addd
aa
ccd
add
ad

-----Sample Output:-----
7
user: Chang's new maths teacher is very enthusiastic about making sure that students understand the concept rather than remembering it. On her first day at teaching, she gives an assignment to all the students to test their potential. Chang wants to demonstrate that he understands the concept rather than rote learning. Help Chang in doing this by solving the hardest problem in the assignment. The problem is written as follows. 

A perfect function is defined in the following manner.

- F(x, y) = x2 + y for 1 ≤ x ≤ A, 1 ≤ y ≤ B
- F(x, y) = 2 otherwise

Find the number of integral pairs (x, y) such that F(x, y) is a perfect square.

-----Input-----
First and the only line of the input contains two single space separated integers A and B.

-----Output-----
Output a single integer indicating the output to the only test case.

-----Constraints-----
- 1 ≤ A, B ≤ 106

-----Example-----
Input:
4 4

Output:
1

-----Explanation-----
Example case 1.
(1, 3) is the only pair such that x = 1, y = 3 satisfying x <= 4 and y <= 4. 1^2 + 3 = 4 which is a perfect square.
user: Fizza is a cute girl and she wants to be more beautiful.
So she approached a magician to increase her beauty magically. But the magician's 
glasses accidently got locked inside the locker and he forgot the password. Without
glasses he can't help her. The password is hidden in N integers i.e. $a[1], a[2], ..... a[N]$ 
written on his magical stick.The password is the maximum number of moves Fizza can perform, for any position $i$ such that 1 < = $i$ < = $N$ and integer $X$ is present at $i$-th
position. In one move she can perform one of the following operations ->
$(1)$  If ( $i$ + 1 < = $N$ ) she can go to position ($i$ + 1 ) if integer at ( $i$ + 1 )-th 
position is equal to $X$.
$(2)$ if ( $i$ + 2 < = $N$ ) she can go to position ($i$ + 2 ) if integer at ($i$ + 2)-th 
position is equal to $X$.
Fizza is a bit worried about her beauty so she can't focus on the problem but she asked for 
your help to find the password.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase consists of  two lines of input.
- Input $N$.
- Input $N$ integers . 

-----Output:-----
For each testcase, output in a single line Maximum number of moves.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq N \leq 10^5$
- $1 \leq a[i] \leq 2*10^5$

-----Sample Input:-----
3
8
6 3 6 4 5 4 3 6
9
5 5 4 5 2 1 3 4 2
6
1 2 3 4 5 6

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

-----EXPLANATION:-----
In the first case, integers at position 1 and 3 are the same and at alternate positions.
In the second case, integers at position 1, 2, 4 follow the conditions.
user: A repetition-free number is one in which each digit $\{1,2,3,…,9\}$ appears at most once and the digit $0$ does not appear. A repetition-free number can have at most nine digits, but may also have fewer than nine digits. Some examples of repetition-free numbers are $9, 32, 489, 98761$ and $983245$.
You will be given an integer $N$ with at most nine digits. Your task is to print out the smallest repetition-free number bigger than $N$. For example, for $99$ the answer is $123$, for $881$ the answer is $891$, and for $133$ the answer is $134$.

-----Input:-----
A single line with a single integer with at most $9$ digits.

-----Output:-----
A single line containing the smallest repetition-free number bigger than the given number. If there is no repetition-free number bigger than the given number, print $0$.

-----Constraints:-----
- $N$ consists of atmost $9$ digits

-----Sample input-----
99

-----Sample output-----
123
user: IIST is thinking of acquiring some land nearby to build its new state of the art labs. The land it has chosen incidentaly has some abandoned college buildings which IIST wants to use. The administration decide the value of the building based on the amount of work that has to be done to get it in shape (lower amount of work, higher value). The security experts want to tear down some of the buildings and use the bricks to build a boundary wall. 

The director has chosen you to write a program which determines the buildings which needs to be demolished to maximise the value while fulfilling the above criterion. 

Input Format: 

The first line contains the number of test cases, T.

The next T cases contains information about hypothetical plots.

Each test case starts with a single integer n, 2 ≤ n ≤ 15, the number of abandoned buildings in the plot. The buildings are identified by consecutive integers 1 to n. Each of the subsequent lines contains 4 integers x_i , y_i , v_i , l_i that describe a single building. (x_i, y_i) is the position of the building in the plane, v_i is its value, and l_i is the length of boundary wall that can be built using the bricks from the building. v_i and l_i are between 0 and 10,000.

Output Format:

For each test case, compute a subset of the buildings such that, using the bricks from the buildings from that subset, the remaining buildings can be enclosed in a single boundary. Find the subset with a minimum value. If more than one such minimum-value subset exists, choose one with the smallest number of buildings. 

Display, as shown below, the identity of each building to be demolished, and the length of the excess boundary (accurate to two fractional digits).

Sample Input:  

2

6

0 0 8 3

1 4 3 2

2 1 7 1

4 1 2 3

3 5 4 6

2 3 9 8

3

3 0 10 2

5 5 20 25

7 -3 30 32

Sample Output: 

2 4 5

3.16

2

15.00
user: Chef has a string of size $N$ which consists only lowercase English alphabet. The chef doesn't like the consonant alphabet at all. So he is thinking of changing every single consonant alphabet to any vowel alphabet. There is some cost for performing this operation.
- Number all alphabet [a,b,c,……,z] as [1,2,3,…..,26]
- So if you want to change c to e then cost will be |e-c| = |5-3| = 2
You need the answer at what minimum cost chef can change every single consonant alphabet to any vowel alphabet. 

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains of a single line of input, a string of lowercase alphabet. 

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

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq |s| \leq 10^2$

-----Sample Input:-----
2
aeiou
dbcc  

-----Sample Output:-----
0
6

-----EXPLANATION:-----
In the first test case, all characters are already vowel so we don't need to change.
In the second tect case
|e-d|=|5-4|=1
|a-b|=|1-2|=1
|a-c|=|1-3|=2
|a-c|=|1-3|=2
1+1+2+2=6
user: Chef has  $N$ doggo (dogs) , Lets number them $1$ to $N$.

Chef decided to build houses for each, but he soon realizes that keeping so many dogs at one place may be messy. So he decided to divide them into several groups called doggo communities. Let the total no. of groups be $K$ . In a community, paths between pairs of houses can be made so that doggo can play with each other. But there cannot be a path between two houses of different communities for sure. Chef wanted to make maximum no. of paths such that the total path is not greater then $K$.
Let’s visualize this problem in an engineer's way :)
The problem is to design a graph with max edge possible such that the total no. of edges should not be greater than the total no. of connected components.

-----INPUT FORMAT-----
- First line of each test case file contain $T$ , denoting total number of test cases.
- $ith$ test case contain only one line with a single integer $N$ , denoting the number of dogs(vertex)

-----OUTPUT FORMAT-----
- For each test case print a line with a integer , denoting the maximum possible path possible.

-----Constraints-----
- $1 \leq T \leq 10000$
- $1 \leq N \leq 10^9$ 

-----Sub-Tasks-----
- $20 Points$
- $1 \leq N \leq 10^6$ 

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

4

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

-----Explanation-----
4 houses can be made with  like this:

community #1  : [1 - 2  ]

community #2 :  [3 - 4 ]

or  [1 - 2 - 3]  , [ 4 ]

In both cases the maximum possible path is 2.
user: Chef has an old machine if the chef enters any natural number, the machine will display 1, 2, …n, n-1, n-2, n-3,…1 series and in next line prints sum of cubes of each number in the series. Chef wants to create a computer program which can replicate the functionality of the machine. Help the chef to code.

-----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-----
- $1 \leq T \leq 50$
- $1 \leq N \leq 50$

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

-----Sample Output:-----
1
45

-----EXPLANATION:-----
For 2) series will be 1, 2, 3, 2, 1 and the sum will be = 1 + 8 + 27 + 8+ 1
user: -----Problem Statement-----
Levy's conjecture, named after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. To put it algebraically, 2n + 1 = p + 2q always has a solution in primes p and q (not necessary to be distinct) for n > 2. (Source: Wikipedia)
In this problem, given a positive integer N (not necessary to be odd integer greater than 5). Your task is to calculate how many distinct ordered pairs (p, q) such that N = p + 2q, where p and q are primes.

-----Input-----
The first line of input contains an integer T, denoting the number of test cases. Then T test cases follow.

Each test case consists of exactly one line containing an integer N.

-----Constraints-----
- 1 ≤ T ≤ 100000 (105)
- 1 ≤ N ≤ 10000 (104)

-----Output-----
For each test case, output the number of ordered pairs (p, q) of primes such that N = p + 2q.

-----Example-----
Input:
3
2
7
11

Output:
0
1
2

-----Explanation-----
Case #1: There are no ordered pairs (p, q) such that p + 2q = 2.
Case #2: There is only one ordered pair (p, q) = (3, 2) such that p + 2q = 7.
Case #3: There are two ordered pairs (p, q) = (7, 2), (5, 3) such that p + 2q = 11.
user: You are given a convex polygon $P$ with vertices $P_0, P_1, \ldots, P_{n-1}$, each having integer coordinates. On each edge $P_{i} P_{(i+1) \% n}$ of the polygon, choose a point $R_i$ uniformly at random. What is the expected area of the convex hull of these $n$ chosen points $R_0, R_1, \ldots R_{n-1}$ ?

-----Note-----
- Consider the area of the convex hull as zero if it contains less than 3 vertices.
- All the points $R_i$ are chosen independently of each other.
- Your answer is considered correct if and only if its absolute or relative error doesn't exceed $10^{-6}$.

-----Input-----
- The first line contains $n$, the number of vertices in the convex polygon.
- The next $n$ lines contain the coordinates of the vertices of the polygon in anti-clockwise order. 

-----Output-----
For each testcase, print the expected area of the convex hull of the $n$ randomly chosen points.

-----Constraints-----
- $3 \leq n \leq 10^5$
- The absolute values of all the coordinates $\leq 10^7$.
- All the points in the input are distinct.
- The described polygon $P$ is convex and the vertices of the polygon are given in anti-clockwise order. Also, no three vertices of the polygon are collinear.

-----Example Input-----
3
0 0
1 0
0 1

-----Example Output-----
0.1250000000
user: Tired of the overpopulated world, Miu - The introverted cat visits a new continent in search for a new house.  
There are $N$ houses lying on the X-axis.

Their positions are given by $X$$i$ , where $i$ refers to the $i$th house. ( $1 <= i <= N$ )

Each of these positions are pairwise distinct 
Miu is supposed to choose one of these houses for herself.

Miu defines a term - The Distance of Peace,  as the minimum distance from her house to any other house.  
Miu wonders what is maximum Distance of Peace she can obtain. Can you help her?

-----Input:-----
- The first line of the input consists of a single integer $T$, denoting the number of test cases  
- The first line of each test case consists of a single integer $N$ 
- The second line of each test case consists of $N$ space-separated integers
$X$$1$ $X$$2$ $X$$3$ … $X$$N$

-----Output:-----
- For each test case print the answer in a single line, the maximum Distance of Peace Miu can obtain

-----Constraints-----
- 1 <= $T$ <= 100  
- 2 <= $N$ <= 105 
- -109 <= $X$$i$ <= 109 
- Sum of $N$ over all test cases does not exceed 106 

-----Subtasks-----
Subtask #1 (30 points): 
- $N$ <= 103 
Subtask #2 (70 points): 
- Original Constraints

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

6

7 -1 2 13 -5 15

4

6 10 3 12  

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

3  

-----EXPLANATION:-----
Test Case 1:

The $1$st house has the maximum Distance of Peace, which is from the $3$rd house:
$| 7 - 2 | = 5$

Hence, the answer is $5$
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-----
- $1 \leq T \leq 50$
- $1 \leq K \leq 50$

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

-----Sample Output:-----
*
*
***
*
* *
*****
*
* *
*   *
*******
*
* *
*   *
*     *
*********   

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: You are given an array with $N$ integers: $A[1], A[2], \ldots, A[N]$ (where $N$ is even). You are allowed to permute the elements however you want. Say, after permuting the elements, you end up with the array $A'[1], A'[2], \ldots, A'[N]$. Your goal is to maximize the following sum:
|A′[1]−A′[2]| + |A′[3]−A′[4]| + ... + |A′[N−1]−A′[N]||A′[1]−A′[2]| + |A′[3]−A′[4]| + ... + |A′[N−1]−A′[N]|
|A'[1] - A'[2]| \ + \ |A'[3] - A'[4]| \ + \ ... \ + \ |A'[N - 1] - A'[N]|

Here, $|x|$ denotes the absolute value of $x$.
You have to print the maximum sum achievable.

-----Input-----
- The first line contains $T$, the number of test cases.
- Each test case starts with an integer $N$ in the first line.
- The second line of each test case contains $N$ space separated integers, denoting the values of array $A$.

-----Output-----
For each test case, output the maximum sum achievable in a new line.

-----Constraints-----
- $1 \le T \le 10^5$
- $1 \le N \le 10^5$
- $N$ is even
- $|A[i]| \le 10^9$
- Sum of $N$ over all test cases $\le 2 * 10^5$

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

-----Example Output 1-----
9

-----Explanation 1-----
The original array is {$1, -3, 2, -3$}. Suppose you permute it and get the array {$2, 1, -3, -3$}. Then the corresponding sum would be $|2 - 1| \ + \ |-3 - (-3)| = 1 + 0 = 1$.
But suppose you permute it differently and get the array {$-3, 2, 1, -3$}. Then the corresponding sum would be $|-3 - 2| \ + \ |1 - (-3)| = 5 + 4 = 9$. You can check that you cannot do any better, and hence the answer is 9.
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:-----
4
1
2
3
4

-----Sample Output:-----
1 
1 10 
11 100 
1 10 11 
100 101 110 
111 1000 1001 
1 10 11 100 
101 110 111 1000 
1001 1010 1011 1100 
1101 1110 1111 10000 

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: The chef has one array of N natural numbers (might be in sorted order). Cheffina challenges chef to find the total number of inversions in the array.

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains two lines of input, $N$.
- N space-separated natural numbers. 

-----Output:-----
For each test case, output in a single line answer as the total number of inversions.

-----Constraints-----
- $1 \leq T \leq 10$
- $1 \leq N \leq 10^5$
- $1 \leq arr[i] \leq 10^5$

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

-----Sample Output:-----
8
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:-----
3
2
3
4

-----Sample Output:-----
21
1
123
21
1
4321
123
21
1

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: In ChefLand, there is a mountain range consisting of $N$ hills (numbered $1$ through $N$) in a straight line. Let's denote the height of the $i$-th hill from the left by $h_i$.
Ada is working on the water supply system of ChefLand. On some of the hills, she wants to place water reservoirs; then, for each reservoir, she will decide in which direction the water should flow from it — either to the left or to the right (water may not flow in both directions from the same reservoir). From a reservoir on a hill with height $h$, water flows in the chosen direction until it reaches the first hill that is strictly higher than $h$; all hills before this hill (including the hill containing the reservoir) are therefore supplied with water.
For example, suppose we have hills with heights $[7, 2, 3, 5, 8]$. If we place a reservoir on the hill with height $5$, and pump water from it to the left, then the hills with heights $2$, $3$ and $5$ are supplied with water.
Help Ada find the minimum numer of reservoirs needed to provide water to all the hills if she chooses the directions optimally.

-----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 $h_1, h_2, \dots, h_N$.

-----Output-----
For each test case, print a single line containing one integer — the minimum required number of reservoirs.

-----Constraints-----
- $2 \le N \le 10^5$
- $1 \le h_i \le 10^9$ for each valid $i$
- $h_i \neq h_j $ for any valid $i \neq j$
- the sum of $N$ over all test cases does not exceed $5 \cdot 10^5$

-----Example Input-----
1
6
4 16 32 6 8 2

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

-----Explanation-----
Example case 1: We can place reservoirs on the second and third hill, pumping water to the left and right respectively.
user: -----Problem-----

There is an infinite one dimensional array ranging from (-infinity, infinity).A Zombie is currently at cell number 0. The zombie wants to reach cell number H. The Zombie moves in only two ways. The Zombie either

Makes U steps to the right (positive side) or

Makes D steps to the left (negative side).

Your task is to find the minimum number of moves the Zombie require to reach the goal.

-----Input-----

The first line contains the number of test cases T. Each of the next T lines contains 3 space

separated integers, H U D.

-----Output-----

For each test case, output one line with an integer, the minimum number of moves required to reach H from 0. If it is impossible, print -1 instead.

-----Constraints-----

- 

T ≤ 10000

- 

1 ≤ H, U, D ≤ 109

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

3 2 1

3 2 2

-----Sample Output-----
3

-1

-----Explanation-----

- 

In the first sample case, first move 2 steps to the right to reach cell number 2. Then 1 step to the left  to reach cell number 1 and finally 2 more steps to the right to reach the goal. Thus 3 moves are required

which is the minimum.

- 

Because both U and D are even, you will always be in an even cell.Thus there is no way to reach cell      number 3.

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