meetforever/GRPO_CL huggingface dataset:
user: Chef wants to teach a lesson of sharing to the students.
There are $N$ students (numbered from $1$ to $N$ from left to right) who are asked to stand in a row. Initially Chef gave $A$$i$ candies to the $i$$th$ child. In one operation any child can give any number of candies to the child standing to his immediate left (i.e. $i$$th$ child can give any amount of candies to the $(i-1)$$th$ child. In particular 1st child cannot give his candies to anyone).  
He asked them to minimize the maximum value of candies a student can possess after performing any number of operations (possibly zero). 
Help the students finding such maximum value.

-----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.
- First line of each test case contains a single integer $N$ denoting the number of students.
- Second line contains $N$ space-separated integers $A$$1$,$A$$2$,$.....$ $A$$N$ denoting the initial amount of candies chef gave to them.

-----Output:-----
- For each test case, print a single line containing one integer ― maximum value after sharing.

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq N \leq 10^5$
- $0$ $\leq$ $A$$i$ $\leq$ $10^9$
- Sum of $N$ over all Test Cases does not exceed $10^5$

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

-----Sample Output-----
3 
5  

-----Explanation-----
- 
For First Test Case:
The $5$$th$ student will give $2$ candies to $4$$th$ student and $4$$th$ will give $3$ candies to $3$$rd$ and $3$$rd$ will give $3$ candies to $2$$nd$ and $2$$nd$ will give $2$ candies to $1$$st$. So finally the number of candies that they will have are 
$[3,3,3,3,3]$ and the value of maximum candies is $3$.
- 
For Second Test Case:
Sharing to the left student will not change the maximum value as $1$$st$ cannot share to anyone. So the maximum value will remain $5$.
user: On her way to ChefLand, Marichka noticed $10^K$ road signs (numbered $0$ through $10^K - 1$). For each valid $i$, the sign with number $i$ had the integer $i$ written on one side and $10^K-i-1$ written on the other side.
Now, Marichka is wondering — how many road signs have exactly two distinct decimal digits written on them (on both sides in total)? Since this number may be large, compute it modulo $10^9+7$.
For example, if $K = 3$, the two integers written on the road sign $363$ are $363$ and $636$, and they contain two distinct digits $3$ and $6$, but on the road sign $362$, there are integers $362$ and $637$, which contain four distinct digits — $2$, $3$, $6$ and $7$. On the road sign $11$, there are integers $11$ and $988$, which contain three distinct digits — $1$, $9$ and $8$.

-----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 a single integer $K$.

-----Output-----
For each test case, print a single line containing one integer — the number of road signs with exactly two digits, modulo $10^9+7$.

-----Constraints-----
- $1 \le T \le 10^5$
- $1 \le K \le 10^9$

-----Subtasks-----
Subtask #1 (20 points): $1 \le T, K \le 5$
Subtask #2 (80 points): original constraints

-----Example Input-----
1
1

-----Example Output-----
10
user: Sereja have array A' that contain N integers. Now Sereja want to permute elements of the array, he want to use some permutation p, such that A[i] = A'[p[i]], where A - new array. 

Lets function f(A,i) = S - A[i] - A[i +1] - ... - A[j], where j is the maximum possible index, such that A[i] + A[i + 1] + ... + A[j] <= S, if A[i] > S, f(A, i) = S.

Help Sereja to find such permutation p, such that (f(A, 1) + f(A, 2) + ... f(A, k))/k will be as low as possible. 

-----Input-----

First line of input contain integer T - number of testcases. Next lines contain T testcases. First line of each testcase contain three integers N, k, S. Next line contain N integers - array A'.

-----Output-----

For each testcase output N numbers in one line - permutation p. 

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ N ≤ 2000
- 1 ≤ k ≤ N
- 1 ≤ A'[i] ≤ 10^4
- 1 ≤ S ≤ 10^9

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

Output:
2 1 3
4 3 2 1


-----Scoring-----

Suppose Sum will be sum of yours (f(A, 1) + f(A, 2) + ... f(A, k))/k per each testcase.  
Lets B will be the smallest such sum. Your score will be equal to B/Sum. Lower scores will earn more points.

We have 20 official test files. You must correctly solve all test files to receive OK. During the contest, your overall score is the sum of the scores on the first 4 test files. After the contest, all solutions will be rescored by the sum of the scores on the rest 16 test files. Note, that public part of the tests may not contain some border cases.
user: Tweedle-Dee and Tweedle-Dum are playing a fierce match of binary Nim. This novel game is played with $N$ stacks, each of them containing only $1$-s and $0$-s.
Just like in normal Nim, Tweedle-Dee and Tweedle-Dum alternate turns; in their turn, a player must choose one non-empty stack and remove a positive number of elements from the top of this stack. However, Tweedle-Dee may only choose a stack with $0$ at the top (and remove elements from it afterwards), and similarly, Tweedle-Dum may only choose a stack with $1$ at the top. the player that cannot make a move loses
Suzumo does not want to wait for the end of the game, so given the starting player he asks you to determine the winner. Remember that Tweedle-Dee and Tweedle-Dum are legendary grandmasters of combinatorial games, so both always play 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 an integer $N$, a string $S$ denoting the number of stacks, the player that starts the game, respectively.
- Each of the following $N$ lines contains a single binary string $B$ representing a stack; the first character of $B$ corresponds to the top element of the stack and the last character to the bottom element.

-----Output-----
For each test case, print a single line containing the string "Dee" if Tweedle-Dee wins the match or "Dum" if Tweedle-Dum wins.

-----Constraints-----
- $1 \le T \le 500$
- $1 \le N \le 50$
- $1 \le |B| \le 50$
- each character in $B$ is either '1' or '0'
- $S$ is either "Dee" or "Dum"

-----Example Input-----
2
2 Dee
101
010
2 Dum
101
010

-----Example Output-----
Dum
Dee
user: Kabir likes Tara's smile and wants to look at her smile. They are sitting in the class and you are friends with Kabir. You have to place a mirror (point size)  in the front wall of the class so that Kabir can have a glimpse of Tara's smile.
Consider the front wall as x-axis . You are given the  coordinates of position of Kabir (x1,y1)$(x1,y1)$ and Tara  (x2,y2)$(x2,y2)$. Find the position where the mirror should be placed. 

-----Input:-----
- First line will contain T$T$, number of testcases. Then the testcases follow. 
- First line of each testcase contains two integers x1,y1$x1, y1$.
- Second line of each testcase contains two integers x2,y2$x2, y2$. 

-----Output:-----
For each testcase, print the x-coordinate of the mirror. Absolute error of 10−2$10^{−2}$ is allowed.

-----Constraints:-----
- 1≤T≤1000$1 \leq T \leq 1000$
- 1≤x1,y1,x2,y2≤105$1 \leq x1,y1,x2,y2 \leq 10^5 $

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

4 4

-----Sample Output:-----
1.60
user: The most dangerous cyborg Jenish is finally entrapped on a narrow bridge over a valley. The bridge is $N$ meters long. For convenience, position $i$ represents a portion of bridge between whose left border is at distance $i-1$ meters from left end and right border is at distance $i$ meters from left end. 
There are $N$ batteries placed at positions $1,2...N$. The $i^{th}$ one has energy $a[i]$.
There are two tanks of strength $X$ at positions $0$ and $N+1$. Initially Jenish has $0$ energy.
From now at any second $t$ starting from $1$, 
first, Jenish can select any battery placed at position $i$ such that $(t \leq i \leq N-t+1)$ and add $a[i]$ to his energy (every battery can be used atmost once).
Then both the tanks move one meter towards each other. If there are still any batteries present at positions where the tanks are heading, the battery gets destroyed. 
At any moment if Jenish's total energy is greater than or equal to $X$, he destroys both the tanks and he escapes the cops. If by the end of $\lfloor \frac {(n+1)}{2}\rfloor^{th}$ second, he can't destroy the tanks, he himself gets destroyed. Find out if he can escape.

-----Input:-----
- The first line consists of a single integer $T$, the number of test cases.
- The first line of each test case contains two space separated integers which represents $N$ and $X$ for that test case respectively.
- The second line of each test case contains $N$ space separated integers, $i^{th}$ of which represents $a[i]$.

-----Output:-----
For each test case, print in a single line, $YES$ if Jenish can escape or $NO$ if he gets destroyed.

-----Constraints:-----
- $1 \leq  X \leq 10^9$
- $0 \leq a[i] \leq 10^6$
- $ 1 \leq N \leq 10^5$
- $\Sigma$ $N$ over all the test cases does not exceed $10^5$

-----Sample Input:-----
3
4 8
5 1 4 2
3 4
3 1 2
2 7
5 5

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

-----Explanation-----
For test $1$, in the 1st second, first Jenish will select battery at index $1$ and then the tanks from either side will move one meter closer, thus destroying the battery at index $4$.Then, in the next second, Jenish will select battery at index $3$.Then,tanks will move one meter closer again to destroy the remaining battery at index $2$.But now, Jenish has a total energy of $9$ units which is more than enough to destroy the tanks.
For test $2$, Jenish can use batteries at index $1$ and $2$ to get a total energy of $4$ units.
For test $3$, Jenish can use batteries at index $1$ or $2$ and get a maximum of $5$ units of energy which is less than required.
user: Once upon a time, a king and a few of his soldiers were  caught by an enemy king in a war. 

He puts them in a circle. The first man in the circle has to kill the second man, the third man has to kill the fourth, fifth man to kill the sixth and so on. When the circle is completed, the remaining people have to form a circle and the process has to repeat. The last man standing will be set free.

If the king has to be set free, which position must he take? For any given N number of people, write a program to find the position that the king has to take.

-----Input-----
Any positive integer in the range 1 to 10000.

-----Output-----
A positive integer indicating safest position

-----Example-----
Input:
9

Output:
3
user: Chef has come to a 2 dimensional garden in which there are N points. Each point has coordinates (x, y), where x can either be 1 or 2 or 3. Chef will now choose every triplet of these N points and make a triangle from it. You need to tell the sum of areas of all the triangles the Chef makes.
Note that some of the triplets might not form proper triangles, and would end up as a line or a point (ie. degenerate), but that is fine because their area will be zero.

-----Input-----
- The first line contains a single integer T, the number of test cases. The description of each testcase follows.
- The first line of each test case contains an integer N denoting the number of points on the plane.
- The next N lines contain 2 space separated integers x and y denoting the coordinates of the points. 

-----Output-----
For each test case, output a single line containing the answer. Your answer will be considered correct if the absolute error is less than or equal to 10-2.

-----Constraints-----
- 1 ≤ T ≤ 20
- 1 ≤ N ≤ 2000
- 1 ≤ x ≤ 3
- 1 ≤ y ≤106
- All (x, y) pairs are distinct

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

Output:
1.0
2.0

-----Explanation:-----
Test Case 1: There is only one triangle which has non-zero area, and it's area is 1, hence the output.
Test Case 2: Let the points be A(1,1), B(2,2), C(2,1), D(3,3). There are 3 non degenerate triangles possible. 

-  area ABC = 0.5
-  area BCD = 0.5
-  area ACD = 1
Total area = 2
user: The annual snake festival is upon us, and all the snakes of the kingdom have gathered to participate in the procession. Chef has been tasked with reporting on the procession, and for this he decides to first keep track of all the snakes. When he sees a snake first, it'll be its Head, and hence he will mark a 'H'. The snakes are long, and when he sees the snake finally slither away, he'll mark a 'T' to denote its tail. In the time in between, when the snake is moving past him, or the time between one snake and the next snake, he marks with '.'s.
Because the snakes come in a procession, and one by one, a valid report would be something like "..H..T...HTH....T.", or "...", or "HT", whereas "T...H..H.T", "H..T..H", "H..H..T..T" would be invalid reports (See explanations at the bottom).
Formally, a snake is represented by a 'H' followed by some (possibly zero) '.'s, and then a 'T'. A valid report is one such that it begins with a (possibly zero length) string of '.'s, and then some (possibly zero) snakes between which there can be some '.'s, and then finally ends with some (possibly zero) '.'s.

Chef had binged on the festival food and had been very drowsy. So his report might be invalid. You need to help him find out if his report is valid or not.

-----Input-----
- The first line contains a single integer, R, which denotes the number of reports to be checked. The description of each report follows after this.
- The first line of each report contains a single integer, L, the length of that report.
- The second line of each report contains a string of length L. The string contains only the characters '.', 'H', and 'T'.

-----Output-----
- For each report, output the string "Valid" or "Invalid" in a new line, depending on whether it was a valid report or not.

-----Constraints-----
- 1 ≤ R ≤ 500
- 1 ≤ length of each report ≤ 500

-----Example-----
Input:
6
18
..H..T...HTH....T.
3
...
10
H..H..T..T
2
HT
11
.T...H..H.T
7
H..T..H

Output:
Valid
Valid
Invalid
Valid
Invalid
Invalid

-----Explanation-----
"H..H..T..T" is invalid because the second snake starts before the first snake ends, which is not allowed.
".T...H..H.T" is invalid because it has a 'T' before a 'H'. A tail can come only after its head.
"H..T..H" is invalid because the last 'H' does not have a corresponding 'T'.
user: Chef belongs to a very rich family which owns many gold mines. Today, he brought N gold coins and decided to form a triangle using these coins. Isn't it strange?
Chef has a unusual way of forming a triangle using gold coins, which is described as follows:
- He puts 1 coin in the 1st row.
- then puts 2 coins in the 2nd row.
- then puts 3 coins in the 3rd row.
-  and so on as shown in the given figure.
Chef is interested in forming a triangle with maximum possible height using at most N coins. Can you tell him the maximum possible height of the triangle?

-----Input-----
The first line of input contains a single integer T denoting the number of test cases. 
The first and the only line of each test case contains an integer N denoting the number of gold coins Chef has.

-----Output-----
For each test case, output a single line containing an integer corresponding to the maximum possible height of the triangle that Chef can get.

-----Constraints-----
- 1 ≤ T ≤ 100
- 1 ≤ N ≤ 109

-----Subtasks-----
- Subtask 1 (48 points) : 1 ≤ N ≤ 105
- Subtask 2 (52 points) : 1 ≤ N ≤ 109

-----Example-----
Input3
3
5
7

Output2
2
3

-----Explanation-----
- Test 1: Chef can't form a triangle with height > 2 as it requires atleast 6 gold coins.
- Test 2: Chef can't form a triangle with height > 2 as it requires atleast 6 gold coins.
- Test 3: Chef can't form a triangle with height > 3 as it requires atleast 10 gold coins.
user: -----Problem Statement-----
A classroom has several students, half of whom are boys and half of whom are girls. You need to arrange all of them in a line for the morning assembly such that the following conditions are satisfied:
- The students must be in order of non-decreasing height.
- Two boys or two girls must not be adjacent to each other.
You have been given the heights of the boys in the array $b$ and the heights of the girls in the array $g$. Find out whether you can arrange them in an order which satisfies the given conditions. Print "YES" if it is possible, or "NO" if it is not.
For example, let's say there are $n = 3$ boys and $n = 3$ girls, where the boys' heights are $b = [5, 3, 8]$ and the girls' heights are $g = [2, 4, 6]$. These students can be arranged in the order $[g_0, b_1, g_1, b_0, g_2, b_2]$, which is $[2, 3, 4, 5, 6, 8]$. Because this is in order of non-decreasing height, and no two boys or two girls are adjacent to each other, this satisfies the conditions. Therefore, the answer is "YES".

-----Input-----
- The first line contains an integer, $t$, denoting the number of test cases.
- The first line of each test case contains an integer, $n$, denoting the number of boys and girls in the classroom.
- The second line of each test case contains $n$ space separated integers, $b_1,b_2, ... b_n$, denoting the heights of the boys.
- The second line of each test case contains $n$ space separated integers, $g_1,g_2,... g_n$, denoting the heights of the girls.

-----Output-----
Print exactly $t$ lines. In the $i^{th}$ of them, print a single line containing "$YES$" without quotes if it is possible to arrange the students in the $i^{th}$ test case, or "$NO$" without quotes if it is not.

-----Constraints-----
- $1 \leq t \leq 10$
- $1 \leq n \leq 100$
- $1 \leq b_i, g_i \leq 100$

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

-----Sample Output-----
YES

-----EXPLANATION-----
The following arrangement would satisfy the given conditions: $[b_1, g_1, b_2, g_2]$. This is because the boys and girls and separated, and the height is in non-decreasing order.
user: Recently, Chef got obsessed with piano. He is a just a rookie in this stuff and can not move his fingers from one key to other fast enough. He discovered that the best way to train finger speed is to play scales.

There are different kinds of scales which are divided on the basis of their interval patterns. For instance, major scale is defined by pattern T-T-S-T-T-T-S, where ‘T’ stands for a whole tone whereas ‘S’ stands for a semitone. Two semitones make one tone. To understand how they are being played, please refer to the below image of piano’s octave – two consecutive keys differ by one semitone.

If we start playing from first key (note C), then we’ll play all white keys in a row (notes C-D-E-F-G-A-B-C – as you can see C and D differ for a tone as in pattern, and E and F differ for a semitone).

This pattern could be played some number of times (in cycle).


Each time Chef takes some type of a scale and plays using some number of octaves. Sometimes Chef can make up some scales, so please don’t blame him if you find some scale that does not exist in real world.

Formally, you have a set of 12 keys (i.e. one octave) and you have N such sets in a row. So in total, you have 12*N keys. You also have a pattern that consists of letters 'T' and 'S', where 'T' means move forward for two keys (from key x to key x + 2, and 'S' means move forward for one key (from key x to key x + 1).

Now, you can start playing from any of the 12*N keys. In one play, you can repeat the pattern as many times as you want, but you cannot go outside the keyboard.

Repeating pattern means that if, for example, you have pattern STTST, you can play STTST as well as STTSTSTTST, as well as STTSTSTTSTSTTST, as well as any number of repeating. For this pattern, if you choose to repeat it once, if you start at some key x, you'll press keys: x (letter 'S')-> x + 1 (letter 'T')-> x + 3 (letter 'T')-> x + 5 (letter 'S') -> x + 6 (letter 'T')-> x + 8. Also 1 ≤ x, x + 8 ≤ 12*N so as to avoid going off the keyboard.

You are asked to calculate number of different plays that can be performed. Two plays differ if and only if they start at different keys or patterns are repeated different number of times.

-----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 each test case contains scale’s pattern – string s consisting of letters ‘T’ and ‘S’ only.

Second line contains one integer N – number of octaves he’ll be using.

-----Output-----
For each test case output a single number in a line corresponding to number of different scales he’ll play. 

-----Constraints-----
- 1 ≤ T ≤ 105
- 1 ≤ |S| ≤ 100
- 1 ≤ n ≤ 7

-----Subtasks-----
Subtask 1: T < 10 4, N = 1 
Subtask 2: No additional constraints.

-----Example-----
Input:
2 
TTTT
1
TTSTTTS
3

Output:
4
36

-----Explanation-----
Example case 1. In the first case there is only one octave and Chef can play scale (not in cycle each time) starting with notes C, C#, D, D# - four together.
user: The Golomb sequence $G_1, G_2, \ldots$ is a non-decreasing integer sequence such that for each positive integer $n$, $G_n$ is the number of occurrences of $n$ in this sequence. The first few elements of $G$ are $[1, 2, 2, 3, 3, 4, 4, 4, 5, \ldots]$. Do you know the recurrence relation for the Golomb sequence? It is $G_1 = 1$ and $G_{n+1} = 1+G_{n+1-G_{G_n}}$ for each $n \ge 1$. A self-describing sequence, isn't it?
Mr. Strange wants to learn CP, so he asked Chef, who is one of the best competitive programmers in the world, to teach him. Chef decided to test his ability by giving him the following task.
Find the sum of squares of the $L$-th through $R$-th term of the Golomb sequence, i.e. $S = \sum_{i=L}^R G_i^2$. Since the sum can be quite large, compute it modulo $10^9+7$.
Can you help Mr. Strange carry out this task given to him by his teacher?

-----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 $L$ and $R$.

-----Output-----
For each test case, print a single line containing one integer $S$ modulo $10^9+7$.

-----Constraints-----
- $1 \le T \le 10^5$
- $1 \le L \le R \le 10^{10}$

-----Subtasks-----
Subtask #1 (50 points):
- $T \le 10^2$
- $R \le 10^9$
Subtask #2 (50 points): original constraints

-----Example Input-----
3
1 5
2 4
100 100

-----Example Output-----
27
17
441

-----Explanation-----
Example case 1: $1^2 + 2^2 + 2^2 + 3^2 + 3^2 = 27$
Example case 2: $2^2 + 2^2 + 3^2 = 17$
Example case 3: $21^2 = 441$
user: k kids seem to have visited your home for the festival. It seems like the kids
had all been fighting with each other, so you decided to keep them as far as
possible from each other. You had placed n chairs on the positive number line,
each at position x i , 1 ≤ i ≤ n. You can make the kids sit in any of the chairs.
Now you want to know the largest possible minimum distance between each kid.

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each testcase contains two lines. First line contains two space separated integers n and k. Second line contains n space separated values, x1, x2, x3, … ,xn.

-----Output:-----
For each test case print the largest possible minimum distance.

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

2 2

1 2    

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

-----Constraints-----
- $2 \leq n \leq 100000$
- $0 \leq xi \leq 10^9$
- $k \leq n $