meetforever/GRPO_CL huggingface dataset:
user: The game of billiards involves two players knocking 3 balls around
on a green baize table. Well, there is more to it, but for our
purposes this is sufficient.
The game consists of several rounds and in each round both players
obtain a score, based on how well they played. Once all the rounds
have been played, the total score of each player is determined by
adding up the scores in all the rounds and the player with the higher
total score is declared the winner.
The Siruseri Sports Club organises an annual billiards game where
the top two players of Siruseri play against each other. The Manager
of Siruseri Sports Club decided to add his own twist to the game by
changing the rules for determining the winner. In his version, at the
end of each round, the cumulative score for each player is calculated, and the leader and her current lead are found. Once
all the rounds are over the player who had the maximum lead at the
end of any round in the game is declared the winner.

Consider the following score sheet for a game with 5 rounds:
RoundPlayer 1Player 2114082289134390110411210658890

The total scores of both players, the leader and the lead after
each round for this game is given below:RoundPlayer 1Player 2LeaderLead114082Player 1582229216Player 1133319326Player 274431432Player 215519522Player 23
Note that the above table contains the cumulative scores.
The winner of this game is Player 1 as he had the maximum lead (58
at the end of round 1) during the game.
Your task is to help the Manager find the winner and the winning
lead. You may assume that the scores will be such that there will
always be a single winner.  That is, there are no ties.
Input
The first line of the input will contain a single integer N (N
≤ 10000) indicating the number of rounds in the game.  Lines
2,3,...,N+1 describe the scores of the two players in the N rounds.
Line i+1 contains two integer Si and Ti, the scores of the Player 1
and 2 respectively, in round i.  You may assume that 1 ≤ Si ≤
1000 and 1 ≤ Ti ≤ 1000.  
Output
Your output must consist of a single line containing two integers
W and L, where W is 1 or 2 and indicates the winner and L is the
maximum lead attained by the winner.
Example
Input:
5
140 82
89 134
90 110
112 106
88 90

Output:
1 58
user: Dr. S. De teaches computer architecture in NIT Patna. Whenever he comes across any good question(with complexity $k$), he gives that question to students within roll number range $i$ and $j$
At the start of semester he assigns score of $10$ to every student in his class if a student submits a question of complexity $k$, his score gets multiplied by $k$
This month he gave $M$ questions and he is wondering what will be mean of maximum scores of all the student. He is busy in improving his finger print attendance module, can you help him?
Input file may be large so try to use fast input output

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase contains of a First line of input, two integers $N, M$ i.e. Number of students in the class and number of questions given in this month. 
- Next $M$ lines contains 3 integers -$i, j, k$ i.e. starting roll number, end roll number and complexity of the question

-----Output:-----
For each testcase, output in a single line answer - $floor$ value of Mean of maximum possible score for all students.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq N, M \leq 10^5$
- $1 \leq i \leq j \leq N$
- $1 \leq k \leq 100$

-----Subtasks-----
Subtask1 
- 
$1 \leq T \leq 10$
- 
$1 \leq N, M \leq 10^4$
Subtask2
- 
Original Constraints 

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

-----Sample Output:-----
202

-----EXPLANATION:-----
Initial score of students will be : $[10, 10, 10, 10, 10]$
after solving question 1 scores will be: $[50, 50, 50, 10, 10]$
after solving question 2 scores will be: $[50, 100, 100, 20, 20]$
after solving question 1 scores will be: $[50, 100, 700, 140, 20]$
Hence after all questions mean of maximum scores will $(50+100+700+140+20)/5 = 202$
user: You are given an N × N grid initially filled by zeros. Let the rows and columns of the grid be numbered from 1 to N, inclusive. There are two types of operations can be applied to the grid:

- RowAdd R X: all numbers in the row R should be increased by X.
- ColAdd C X: all numbers in the column C should be increased by X.

Now after performing the sequence of such operations you need to find the maximum element in the grid.

-----Input-----
The first line of the input contains two space separated integers N and Q denoting the size of the grid and the number of performed operations respectively. Each of the following Q lines describe an operation in the format described above.

-----Output-----
Output a single line containing the maximum number at the grid after performing all the operations.

-----Constraints-----
- 1 ≤ N ≤ 314159
- 1 ≤ Q ≤ 314159
- 1 ≤ X ≤ 3141
- 1 ≤ R, C ≤ N

-----Example-----
Input:
2 4
RowAdd 1 3
ColAdd 2 1
ColAdd 1 4
RowAdd 2 1

Output:
7

-----Explanation-----
The grid changes as follows:
00  33  34  74  74
00  00  01  41  52

The maximum number in the final grid is 7.
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-----
- $1 \leq T \leq 26$
- $1 \leq K \leq 26$

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

-----Sample Output:-----
A
12
A
12
ABC
1234

-----EXPLANATION:-----
No need, else pattern can be decode easily.
user: The activity of a panipuri seller is "making a panipuri and putting it on the palte of his customer".    
$N$ customers are eating panipuri, Given an array $A$ of length $N$, $i^{th}$ customer takes $A_i$ seconds to eat a panipuri.     
The Speed of Panipuri seller refers to the number of customers served per second. Determine the minimum speed of panipuri seller so that no customer has to wait for panipuri after getting his/her first panipuri.    
Assume that the plate can hold infinite panipuris, and customer starts eating next panipuri just after finishing the current one.    You would be provided with the time taken by each customer to eat a panipuri.
Panpuri seller serves panipuri in round robin manner (first to last and then again first).

-----Input:-----
- First line will contain $T$, number of testcases. Then the test cases follow.
- For each test case, the first line contains $N$ number of customers.
- Then the second line will contain $N$ space separated integers, $A_1$ to $A_N$, eating time taken by each customer(in seconds).

-----Output:-----
- For each test case, print a single line containing the minimum integral speed of panipuri seller so that no customer has to wait. 

-----Constraints-----
- $0 < T \leq 100$
- $0 < N \leq 10^6$
- $0 < A_i \leq 10^9$
- Sum of $N$ over all test cases does not exceed $10^6$

-----Sample Input:-----
2
4
2 4 6 3
5
2 3 4 6 5

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

3

-----EXPLANATION:-----
$Test case$ $1$:

Let's assume speed of panipuri seller be 1.

So starting from $0^{th}$ second, in $0$ to $1$ second seller make a panipuri and put it into the plate of first customer, and first customer completely eat it in $1$ to $3$ second as he takes $2$ second to eat a panipuri. But in $3$ to $4$ second panipuri seller is serving to $4^{th}$ customer, and hence first customer have to wait, so 1 cannot be our answer.

Next possible speed can be $2$ .

Starting from $0^{th}$ second, at half second seller make a panipuri and put it into the plate of first customer,  first customer completely eat it on $2.5$ second as he takes $2$ second to eat a panipuri. In $0.5$ to $1$ second, seller serve the second customer. similarly in $1$ to $2$ second $3^{rd}$ and $4^{th}$ customer will be serve, in $2$ to $2.5$ second, seller will again serve the first customer so at $2.5$ sec $1^{st}$ customer will have a panipuri to eat.

Similarly proceeding further we will find that no customer have to wait when speed is $2$.
user: Chef wants to buy a new phone, but he is not willing to spend a lot of money. Instead, he checks the price of his chosen model everyday and waits for the price to drop to an acceptable value. So far, he has observed the price for $N$ days (numbere $1$ through $N$); for each valid $i$, the price on the $i$-th day was $P_i$ dollars.
On each day, Chef considers the price of the phone to be good if it is strictly smaller than all the prices he has observed during the previous five days. If there is no record of the price on some of the previous five days (because Chef has not started checking the price on that day yet), then Chef simply ignores that previous day ― we could say that he considers the price on that day to be infinite.
Now, Chef is wondering ― on how many days has he considered the price to be good? Find the number of these days.

-----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 $P_1, P_2, \dots, P_N$.

-----Output-----
For each test case, print a single line containing one integer ― the number of days with a good price.

-----Constraints-----
- $1 \le T \le 100$
- $7 \le N \le 100$
- $350 \le P_i \le 750$ for each valid $i$

-----Subtasks-----
Subtask #1 (30 points): $N = 7$
Subtask #2 (70 points): original constraints

-----Example Input-----
1
7
375 750 723 662 647 656 619

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

-----Explanation-----
Example case 1: Chef considers the price to be good on day $1$, because he has not observed any prices on the previous days. The prices on days $2, 3, 4, 5, 6$ are not considered good because they are greater than the price on day $1$. Finally, the price on day $7$ is considered good because it is smaller than all of the prices on days $2, 3, 4, 5, 6$.
user: Chef works in a similar way to a travelling salesman ― he always travels to new cities in order to sell his delicious dishes.
Today, Chef is planning to visit $N$ cities (numbered $1$ through $N$). There is a direct way to travel between each pair of cities. Each city has a specific temperature; let's denote the temperature in the $i$-th city by $C_i$. Chef has a fixed temperature tolerance $D$ with the following meaning: for each pair of cities $a$ and $b$, he may travel from city $a$ directly to city $b$ only if $|C_a-C_b| \le D$, otherwise he would catch a heavy flu because of the sudden change in temperature.
Chef starts from city $1$. Is he able to visit all $N$ cities in such a way that each city is visited exactly once?
Notes:
- Chef is not able to travel through a city without visiting it.
- City $1$ is visited at the beginning.
- It is not necessary to be able to travel directly to city $1$ from the last city Chef visits.

-----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 $D$.
- The second line contains $N$ space-separated integers $C_1, C_2, \ldots, C_N$.

-----Output-----
For each test case, print a single line containing the string "YES" (without quotes) if Chef can visit all cities or "NO" (without quotes) if he cannot.

-----Constraints-----
- $1 \le T \le 1,000$
- $2 \le N \le 10^5$
- $0 \le D \le 10^9$
- $0 \le C_i \le 10^9$ for each valid $i$
- the sum of $N$ over all test cases does not exceed $10^6$

-----Subtasks-----
Subtask #1 (20 points):
- $N \le 1,000$
- the sum of $N$ over all test cases does not exceed $10,000$
Subtask #2 (80 points): original constraints

-----Example Input-----
2
5 3
3 2 1 4 5
5 4
10 1 3 2 9

-----Example Output-----
YES
NO
user: A group of rebels travelling on a square hoverboard is ambushed by Imperial Stormtroopers.Their big hoverboard is an easy target, so they decide to split the board into smaller square hoverboards so that they can bolt away easily.But they should also make sure they don't get too spread out.Help the rebels split the craft into minimum number of smaller crafts possible.

-----Input-----
A single integer N denoting the side length of the big hoverboard.

-----Output-----
In the first line, output the integer 'k' which is the minimum number of square boards into which the bigger board can be split up.

In the second line, output k space separated integers which denote the sizes of the smaller square hoverboards.This must be in increasing order of sizes.

-----Constraints-----
N ranges from 2 to 50.

-----Example-----
Input:
3

Output:
6
1 1 1 1 1 2

-----Explanation-----
A square of side length 3 can be split into smaller squares in two ways: Either into 9 squares of side1 or 5 squares of side 1 and 1 square of size 2.The second case is the favourable one.
user: Takaki Tono is a Computer Programmer in Tokyo. His boss at work shows him an online puzzle, which if solved would earn the solver a full expense paid trip to Los Angeles, California. Takaki really wants to solve this, as the love of his life, Akari, lives in Los Angeles and he hasn't met her since four years. Upon reading the puzzle he realizes that it is a query based problem. The problem is as follows :-

You are given a Tree T with N nodes numbered from 1 to N, with each node numbered z having a positive integer Az written on it. This integer denotes the value of the node. You have to process Q queries, of the following forms :- 
1) C x y : Report the closest two values in the unique path from x to y i.e compute min(|Ap - Aq|) where p and q are two distinct nodes on the unique path from x to y. 

2) F x y : Report the farthest two values in the unique path from x to y i.e. compute max(|Ap - Aq|) where p and q are two distinct nodes on the unique path from x to y.

It is also mentioned that x is not equal to y in any query and that no two nodes have the same value printed on them. Also, |x| denotes the absolute value of x. 

Takaki is perplexed and requires your help to solve this task? Can you help him out?

-----Input-----
The first line of the input contains an integer N denoting the number of nodes in tree T. 
The second line comprises N space separated integers denoting A, where the i-th integer denotes Ai. 
The next N-1 lines each comprise two space separated integers u and v, denoting that node u and node v
are connected by an edge. It is guaranteed that the final graph will be a connected tree.
The next line contains a single integer Q, denoting number of queries. 
The next Q lines comprise the queries. Each such line is of the format C x y or F x y. 

-----Output-----
For each query, print the required output as mentioned above. 

-----Constraints-----
- 2 ≤ N ≤ 35000
- 1 ≤ Ai ≤ 109
- 1 ≤ Q ≤ 35000
- 1 ≤ u, v ≤ N
- No two nodes have the same value printed on them.
- x is not equal to y in any query.

-----Subtasks-----

-----Subtasks-----Subtask #1 (15 points)
- N, Q ≤ 1000Subtask #2 (20 points)
- Only Type F queries are present.Subtask #3 (65 points)
- Original constraints

-----Example-----
Input:5
1 2 7 4 5
1 2
2 3
2 4
2 5
7
C 1 5
F 1 5
C 2 4
C 1 2
F 1 3
F 3 4
F 2 4

Output:1
4
2
1
6
5
2

-----Explanation-----
Given below is the tree corresponding to the sample input. Each node has two numbers written in it. 
The first number represents the node index and the second number indicates node value.
user: Chef Leonardo has a decimal integer $N$ and a non-zero decimal digit $d$. $N$ does not contain the digit zero; specifically, $N$ should always be treated as a decimal integer without leading zeroes.
Chef likes $d$ and does not like any other digit, so he decided to change $N$. He may apply the following operation any number of times (including zero): append the digit $d$ to the decimal representation of $N$ ($d$ becomes the least significant digit of $N$), then remove one occurrence of one digit from the decimal representation of $N$.
Chef has already changed a lot of numbers and he got bored. Now, he wants to know the smallest possible value of $N$ which can be obtained by performing the operation described above. Can you help him?

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first and only line of each test case contains two space-separated integers $N$ and $d$.

-----Output-----
For each test case, print a single line containing one integer - the minimum number Chef can obtain.

-----Constraints-----
- $1 \le T \le 1,000$
- $1 \le N \le 10^{18}$
- $N$ does not contain the digit $0$
- $1 \le d \le 9$

-----Subtasks-----
Subtask #1 (40 points):
- $1 \le T \le 100$
- $1 \le N \le 10^9$
Subtask #2 (60 points): original constraints

-----Example Input-----
3
35 4
42 4
24 9

-----Example Output-----
34
24
24

-----Explanation-----
Example case 1: Chef can remove the digit $5$ and add $4$ to the end of $N$. Then, $N$ becomes $34$.
Example case 2: Chef can remove the digit $4$ from the beginning of $N$, add $4$ to the end of $N$ and get $N = 24$.
Example case 3: Chef cannot make his number any smaller.
user: Given an array A1,A2...AN, you have to print the size of the largest contiguous subarray such that 
GCD of all integers in that subarray is 1.

Formally,
For a subarray Ai,Ai+1...Aj where 1 ≤ i < j ≤ N to be valid: GCD(Ai,Ai+1...Aj) should be 1. You have to print the size of the largest valid subarray.
If no valid subarray exists, output -1.
Note:A single element is not considered as a subarray according to the definition of this problem.

-----Input-----
First line contains T, the number of testcases. Each testcase consists of N in one line followed by N integers in the next line.

-----Output-----
For each testcase, print the required answer in one line.

-----Constraints-----
- 1 ≤ T ≤ 10
- 2 ≤ N ≤ 105
- 1 ≤ Ai ≤ 105

-----Example-----
Input:
2
2
7 2
3
2 2 4

Output:
2
-1

-----Explanation-----
Example case 1.GCD(2,7)=1. So the subarray [A1,A2] is valid.
Example case 2.No subarray satisfies.
Note: Use scanf/print instead of cin/cout. Large input files.
user: You are given an array A consisting of N integers. A group of disjoint subarrays in it will be a collection of subarrays of the array. Formally a group of subarrays consisting of K subarrays can be denoted by 2 * K indices, [i1, j1], [i2, j2] , ..., [iK, jK], such that i1 ≤ j1 < i2 ≤ j2 < ... < iK ≤ jK.

For example, in array A = {5, 6, 7}. A group of subarrays could be the subarrays denoted by indices [1, 1], [2, 3]. The subarray corresponding to indices [1, 1] will be {5}. The subarray corresponding to indices [2, 3] will be {6, 7}. So, we have i1 = 1, j1 = 1, i2 = 2, j2 = 3 and K = 2. You can check that the indices satisfy the property i1 ≤ j1 < i2 ≤ j2.

Note that the group of subarrays [1, 2] and [2, 3] won't be disjoint, as it does not satisfy the property j1 < i2. In other words, the index 2 is common in two subarrays, which should not happen.

Let M denote the maximum value of K in a group of K disjoint subarrays of array A, such that there are not two elements (not indices) common in those subarrays. This means, that if the group contained subarrays [A[i1], A[j1], [A[i2], A[j2]] , ..., A[[iK, jK]], then there should not be an element which is present in more than one subarrays.

You have to find maximum number of group of disjoint subarrays that number of subarrays in those groups are equal to M. As the answer could be large, output it modulo 109 + 7.

-----Input-----
The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.
First line of the each test case contains a single integer N denoting number of elements in array A.
Second line of each test case contains N space separated integers denoting the elements of the array A

-----Output-----
For each test case, output a single line corresponding to maximum number of group of disjoint subarrays of array A.

-----Constraints-----
- 1 ≤ T, N ≤ 105
- 1 ≤ Ai ≤ n
- Sum of N over all the test cases in a single file won't exceed 105

-----Subtasks-----
Subtask #1 (30 points)

- 1 ≤ T, N ≤ 103
- 1 ≤ Ai ≤ n
- Sum of N over all the test cases in a single file won't exceed 103

Subtask #2 (70 points)
- original constraints

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

Output:
1
3
18

-----Explanation-----
Example case 1. M will be equal to 2. The subarrays will be {[1, 1], [2, 2]}.
Example case 2. M will be equal to 3. The subarrays will be {[1, 1], [2, 2]}, {[1, 1], [3, 3]} and {[1, 1], [2, 3]}. Note that {[2, 2], [3, 3]} won't be a non-intersecting subarray as A[2] = 2 and A[3] = 2. So, 2 is common in both these subarrays.
user: Chef is playing a game on a sequence of $N$ positive integers, say $A_1, A_2, ... A_N$ The game is played as follows.
- If all the numbers are equal, the game ends.
- Otherwise
- Select two numbers which are unequal
- Subtract the smaller number from the larger number
- Replace the larger number with the result from above
Chef has already figured out that the game always terminates. He also knows, for a given sequence of integers, the game will always terminate on the same value, no matter how the game is played. Chef wants you to simulate the game for him and tell him if the game terminates on $1$.
In fact, there may be many such games. Given a sequence of integers Chef wants to know the number of sub-sequences of the given sequence, for which, playing the above game on the subsuquence will terminate on $1$. A sub-sequence can be obtained from the original sequence by deleting $0$ or more integers from the original sequence. See the explanation section for clarity.

-----Input-----
- The first line of the input contains an integer $T$, the number of test cases. Then follow the description of $T$ test cases. 
- The first line of each test case contains a single integer $N$, the length of the sequence. 
- The second line contains $N$ positive integers, each separated by a single space.

-----Output-----
For each test case, output a single integer - the number of sub-sequences of the original sequence, such that, playing the game on the sub-sequence results in ending the game with all the values equal to $1$.

-----Constraints-----
- $1 \le T \le 100$ 
- $1 \le N \le 60$ 
- $1 \le A_i \le 10^4$ 
- All $A_i$ will be distinct.

-----Sample Input-----
3
4
2 3 5 7
4
3 4 8 16
3
6 10 15

-----Sample Output-----
11
7
1

-----Explanation-----
Test Case 1: The following $11$ sub-sequences are counted.
- $\{ 2, 3 \}$
- $\{ 2, 5 \}$
- $\{ 2, 7 \}$
- $\{ 3, 5 \}$
- $\{ 3, 7 \}$
- $\{ 5, 7 \}$
- $\{ 2, 3, 5 \}$
- $\{ 2, 3, 7 \}$
- $\{ 2, 5, 7 \}$
- $\{ 3, 5, 7 \}$
- $\{ 2, 3, 5, 7 \}$
Test Case 2: The following $7$ sub-sequences are counted.
- $\{ 3, 4 \}$
- $\{ 3, 8 \}$
- $\{ 3, 16 \}$
- $\{ 3, 4, 8 \}$
- $\{ 3, 4, 16 \}$
- $\{ 3, 8, 16 \}$
- $\{ 3, 4, 8, 16 \}$
Test Case 3: There are $8$ subsequences of $\{ 6, 10, 15 \}$
- $\{\}$ => The game cannot be played on this sub-sequence
- $\{ 6 \}$ => The game cannot be played on this sub-sequence
- $\{ 10 \}$ => The game cannot be played on this sub-sequence
- $\{ 15 \}$ => The game cannot be played on this sub-sequence
- $\{ 6, 10 \}$ => The game cannot end at $\{ 1, 1 \}$
- $\{ 6, 15 \}$ => The game cannot end at $\{ 1, 1 \}$
- $\{ 10, 15 \}$ => The game cannot end at $\{ 1, 1 \}$
- $\{ 6, 10, 15 \}$ => The game ends at $\{ 1, 1, 1 \}$. Hence this is the only sub-sequence that is counted in the result.