meetforever/GRPO_CL huggingface dataset:
user: A robot is initially at $(0,0)$ on the cartesian plane. It can move in 4 directions - up, down, left, right denoted by letter u, d, l, r respectively. More formally:
- if the position of robot is $(x,y)$ then u makes it $(x,y+1)$
- if the position of robot is $(x,y)$ then l makes it $(x-1,y)$
- if the position of robot is $(x,y)$ then d makes it $(x,y-1)$
- if the position of robot is $(x,y)$ then r makes it $(x+1,y)$
The robot is performing a counter-clockwise spiral movement such that his movement can be represented by the following sequence of moves -
ulddrruuulllddddrrrruuuuu… and so on.
A single move takes 1 sec. You have to find out the position of the robot on the cartesian plane at $t$ second.

-----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 $t$.

-----Output:-----
For each test case, print two space-separated integers, $(x,y)$ — the position of the robot.

-----Constraints-----
- $1 \leq T \leq 10^6$
- $1 \leq t \leq 10^{18}$

-----Sample Input:-----
5
1
2
3
50
12233443

-----Sample Output:-----
0 1
-1 1
-1 0
2 4
-1749 812
user: Prof. Sergio Marquina is a mathematics teacher at the University of Spain. 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 the semester he assigns a 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 students. He is busy planning a tour of the Bank of Spain for his students, can you help him?

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains the 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 contain 3 integers -i,j,k i.e. starting roll number, end roll number, and complexity of the question

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

-----Constraints-----
- $1 \leq T \leq 100$
- $1 \leq N, M \leq 105$
- $1 \leq i \leq j \leq N$
- $1 \leq k \leq 100$

-----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: Taxis of Kharagpur are famous for making sharp turns. You are given the coordinates where a particular taxi was on a 2-D planes at N different moments: (x1, y1), (x2, y2), ..., (xN, yN). In between these coordinates, the taxi moves on a straight line. A turn at the i-th (2 ≤ i ≤ N-1) coordinate is said to be a sharp turn if the angle by which it turns at Point B = (xi, yi) when going from coordinates A = (xi-1, yi-1) to C = (xi+1, yi+1) via (xi, yi) is greater than 45 degrees. ie. suppose you extend the line segment AB further till a point D, then the angle DBC would be greater than 45 degrees.
You have to identify whether the taxi made a sharp turn somewhere or not (Please look at Output section for details). If it made a sharp turn, also identify whether it is possible to change the coordinates at one of the N moments to make sure that the taxi doesn't make any sharp turn. Note that all the N pairs of coordinates (including the new coordinates) should be integers and distinct and should have their x and y coordinates at least 0 and at most 50.

-----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 denoting the number of moments at which you are given the information of the taxi's coordinates.
- Each of the next N lines contains two space-separated integers xi and yi denoting the x and y coordinates of the taxi at i-th moment.

-----Output-----
- For each test case, print a single line containing two space-separated strings, either of which can be a "yes" or "no" (without quotes). If there was no sharp turn, the first string should be "yes", and if there was a sharp turn, the first string should be "no". For the second string, you should tell whether it is possible to modify at most one coordinate in such a way that taxi doesn't make a sharp turn. Note that if the first string is "yes", then the second string would always be "yes".

-----Constraints-----
- 1 ≤ T ≤ 50
- 3 ≤ N ≤ 50
- 0 ≤ xi, yi ≤ 50
- It's guaranteed that all (xi, yi) pairs are distinct.

-----Example-----
Input
5
3
0 0
1 1
2 1
3
0 0
1 0
6 1
3
0 0
1 0
1 1
4
0 0
1 0
1 1
6 1
6
0 0
1 0
1 1
2 1
2 2
3 2

Output
yes yes
yes yes
no yes
no yes
no no

-----Explanation-----
Example 1. 

You can see that taxi is never making a sharp turn.

Example 3

You can see that taxi is making a sharp turn of 90 degrees, an angle greater than 45'. However, you can change the coordinates of the third points to (2, 1) to ensure that the angle remains <= 45'.
user: Chef is the judge of a competition. There are two players participating in this competition — Alice and Bob.
The competition consists of N races. For each i (1 ≤ i ≤ N), Alice finished the i-th race in Ai minutes, while Bob finished it in Bi minutes. The player with the smallest sum of finish times wins. If this total time is the same for Alice and for Bob, a draw is declared.
The rules of the competition allow each player to choose a race which will not be counted towards their total time. That is, Alice may choose an index x and her finish time in the race with this index will be considered zero; similarly, Bob may choose an index y and his finish time in the race with this index will be considered zero. Note that x can be different from y; the index chosen by Alice does not affect Bob's total time or vice versa.
Chef, as the judge, needs to announce the result of the competition. He knows that both Alice and Bob play optimally and will always choose the best option. Please help Chef determine the result!

-----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 A1, A2, ..., AN.
- The third line contains N space-separated integers B1, B2, ..., BN.

-----Output-----
For each test case, print a single line containing the string "Alice" if Alice wins, "Bob" if Bob wins or "Draw" if the result is a draw (without quotes).

-----Constraints-----
- 1 ≤ T ≤ 100
- 2 ≤ N ≤ 100
- 1 ≤ Ai ≤ 1000 for each valid i
- 1 ≤ Bi ≤ 1000 for each valid i

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

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

Output:

Alice
Bob
Draw

-----Explanation-----
Example case 1: Alice will choose the finish time in the last race to be considered zero, which means her sum of finish times is 3 + 1 + 3 + 3 + 0 = 10, while Bob will choose the finish time of his second race to be considered zero, so his total sum of finish times is 1 + 0 + 2 + 5 + 3 = 11. Since Alice's sum is smaller, she is considered the winner.
Example case 2: We're dealing with the same situation as in the previous case, but finish times for the players are swapped, so Bob wins this time.
Example case 3: Alice will choose the finish time of the first race to be considered zero, which means her total time is 0 + 1 + 3 = 4. Bob will choose the finish time of his last race to be considered zero, which makes his total time 2 + 2 + 0 = 4. The competition is considered a draw because both players have equal sums of finish times.
user: This time minions are celebrating Diwali Festival. There are N minions in total. Each of them owns a house. On this Festival, Each of them wants to decorate their house. But none of them have enough money to do that. One of the minion, Kevin, requested Gru for money. Gru agreed for money distribution but he will be giving money to a minion if and only if demanded money is less than or equal to the money Gru have. Now Gru wonders if he can spend all the money or not.

-----Input-----
First line have number of test cases T. Each test case consist of Two Lines. First line contains two space separated integers N and K i.e. Number of minions and Amount of Money Gru have. Next line contains N space separated integers A1,A2,A3,.....,AN representing amount of money demanded by ith minion.

-----Output-----
Output YES if Gru can spend his all of the money on minions i.e. after distribution Gru have zero amount of money else NO.

-----Constraints-----
- 1 ≤ T ≤ 105
- 1 ≤ N ≤ 102
- 1 ≤ K,Ai ≤ 109

-----Example-----
Input:
2 
4 9
5 2 2 4
4 9
5 2 18 3

Output:
YES
NO


-----Explanation-----
Example case 1.At first Gru is having 9 Rs. If he gives 5 Rs. to first minion then remaining 4 Rs. can be given to 2nd and 3rd minion or to the 4th minion. Which will leave zero amount of money in the hands of Gru.
Example case 2.At first Gru is having 9 Rs. If he gives 5 Rs. to the first minion then from remaining 4 Rs. either he can give 2 Rs. to the 2nd minion or 3 Rs. to the fourth minion. Which will leave either 2 Rs. or 1 Rs. in the hands of Gru.
user: Chef loves to play chess, so he bought a new chessboard with width M$M$ and height N$N$ recently.
Chef considers a chessboard correct if its width (number of columns) is equal to its height (number of rows) and each cell has no side-adjacent cell of the same color (this is the so-called "chess order" which you can see in real-world chessboards). Chef's chessboard does not have to be a correct chessboard (in particular, it may have N≠M$N \neq M$).
A sub-board of Chef's chessboard is a rectangular piece of this board with an arbitrarily chosen top left and bottom right cell (possibly equal to the original chessboard). Every sub-board is also a chessboard.
Chef can invert some cells; inverting a cell means changing its color from white to black or from black to white. After inverting those cells, he wants to cut the maximum correct sub-board out of the original chessboard.
Chef has not yet decided how many cells he would like to invert. Now he wonders about the answers to Q$Q$ question. In the i$i$-th question (1≤i≤Q$1 \le i \le Q$), he is allowed to invert at most ci$c_i$ cells (possibly zero); he would like to know the side length of the largest possible correct sub-board of his chessboard. Help Chef answer these questions.

-----Input-----
- The first line of the input contains two space-separated integers N$N$ and M$M$.
- N$N$ lines follow. For each valid i$i$, the i$i$-th of these lines contains a string with length M$M$ describing the i$i$-th row of Chef's chessboard. Each character of this string is either '0', representing a black cell, or '1', representing a white cell.
- The next line contains a single integer Q$Q$.
- The last line contains Q$Q$ space-separated integers c1,c2,…,cQ$c_1, c_2, \dots, c_Q$.

-----Output-----
For each question, print a single line containing one integer — the maximum size of a correct sub-board.

-----Constraints-----
- 1≤N,M≤200$1 \le N, M \le 200$
- 1≤Q≤105$1 \le Q \le 10^5$
- 0≤ci≤109$0 \le c_i \le 10^9$ for each valid i$i$

-----Subtasks-----
Subtask #1 (20 points):
- 1≤N,M≤20$1 \le N, M \le 20$
- 1≤Q≤100$1 \le Q \le 100$
Subtask #2 (30 points): 1≤N,M≤20$1 \le N, M \le 20$
Subtask #3 (50 points): original constraints

-----Example Input-----
8 8
00101010
00010101
10101010
01010101
10101010
01010101
10101010
01010101
4
1 2 0 1001

-----Example Output-----
7
8
6
8

-----Explanation-----
If we don't change the board, the best answer here is the 6x6 bottom right sub-board. We can invert cells (2,2)$(2, 2)$ and (1,1)$(1, 1)$ to get a better answer.
user: The land of Programmers Army is surrounded by many islands. A unique number is associated with each island. The king of the islands is a very generous person, he donates a certain amount of gold coins to travelers for visiting each island that they visited to.
Now, you are appointed as a traveler, who will travel to all these(or some) islands as many times as the Army wants, and you will collect gold coins from the king of the island.
In each trip, you will be asked to give the total sum of gold coins you have collected.

-----Input:-----
- The first line of the input contains a single integer $T$. $T$ denoting the number of test cases. The description of $T$ test cases is as follows.
- The next line of the input contains a single integer $N$. $N$ denotes the total number of Islands.
- The next line of the input contains $N$ space-separated integers $A1, A2, A3...An$ where $ith$ number denotes the maximum number of coins that the king of $ith$ island can donate.
- Next line contains a single integer $Q$. $Q$ denotes the total number of times traveler have to go for the trip.
- Next $Q$ lines contains, two space-separated integers $Q1,Q2$ denoting the start and end number of islands, i.e. traveler will start the trip from $Q1th$ island and will go till $Q2th$ island, in each trip.
Note: islands are numbered from $1$ to $N$.

-----Output:-----
- For each trip print the total number of gold coins, traveler will collect(each on a new line).

-----Constraints:-----
- $1 \leq T \leq 100$
- $1 \leq N \leq 10^4$
- $1 \leq A1, A2, A3...An \leq 10^5$
- $1 \leq Q \leq 10^3$
- $1 \leq Q1,Q2 \leq N$

-----Sample Input:-----
1
4
10 2 5 50
2
1 3
2 4

-----Sample Output:-----
17
57

-----Explanation:-----
- 
In 1st Trip, traveler will go from 1st Island to 3rd Island, hence the total number of coins traveler can collect is 10+2+5 = 17
- 
In 2 d Trip, traveler will go from 2nd Island to 4th Island, hence the total number of coins traveler can collect is 2+5+50 = 57
user: -----Coal Company -----

The Tunisian Coal Mining company uses a train to ferry out coal blocks from its coal mines. The train has N containers numbered from 1 to N which need to

be filled with blocks of coal. Assume there are infinite coal blocks. The containers are arranged in increasing order of capacity, and the ith container

has capacity i. Every container has a specific loading cost ci. The workers fill the containers in rounds. In every round, they choose a subset of

containers and load them with coal blocks. This subset should be such that each subsequent container chosen in a round should be more spacious than the

previous one. Also, the difference in loading cost of consecutive containers should be at least K.



What is the least number of rounds in which all containers can be filled?



-----
Input
-----

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

Each case contains an integer N and K on the first line, followed by integers c1,...,cn on the second line.



1 <= T <= 100

1 <= N <= 300

1 <= ci <= 1000

1 <= K <= 1000



-----
Output
-----

Output T lines, one for each test case, containing the minimum number of rounds in which all containers could be filled.

-----
Example
-----

Input:

2

3 2

5 4 7

5 1

5 3 4 5 6



Output:

2

1







Explanation:

For the first example, workers can fill the containers of cost 5 and 7 in the first round and the container with cost 4 in the next round. Note that the

containers with cost 5 and 4 cannot be filled consecutively because the loading costs should differ by at least K (which is 2). Also, the containers cannot

be filled in order 5, 7, 4 in one round because the containers filled in a round should be in increasing capacity.
user: Chef likes strings a lot but he likes palindromic strings more. Today, Chef has two strings A and B, each consisting of lower case alphabets.

Chef is eager to know whether it is possible to choose some non empty strings s1 and s2 where s1 is a substring of A, s2 is a substring of B such that s1 + s2 is a palindromic string. Here '+' denotes the concatenation between the strings.
Note:
A string is a palindromic string if it can be read same both forward as well as backward. To know more about palindromes click here.

-----Input-----
- First line of input contains a single integer T denoting the number of test cases.
- For each test case:
	
- First line contains the string A
- Second line contains the string B.

-----Output-----
For each test case, Print "Yes" (without quotes) if it possible to choose such strings s1 & s2. Print "No" (without quotes) otherwise.


-----Constraints-----
- 1 ≤ T ≤ 10 
- 1 ≤ |A|, |B| ≤ 1000 

-----Subtasks-----
-  Subtask 1: 1 ≤ |A|, |B| ≤ 10 : ( 40 pts )
-  Subtask 2: 1 ≤ |A|, |B| ≤ 1000 : ( 60 pts )

-----Example-----Input
3
abc
abc
a
b
abba
baab

Output
Yes
No
Yes

-----Explanation-----
- Test 1: One possible way of choosing s1 & s2 is s1 = "ab", s2 = "a" such that s1 + s2 i.e "aba" is a palindrome.
- Test 2: There is no possible way to choose s1 & s2 such that s1 + s2 is a palindrome.
- Test 3: You can figure it out yourself.
user: Chef's new hobby is painting, but he learned the fact that it's not easy to paint 2D pictures in a hard way, after wasting a lot of canvas paper, paint and of course time. From now on, he decided to paint 1D pictures only.
Chef's canvas is N millimeters long and is initially all white. For simplicity, colors will be represented by an integer between 0 and 105. 0 indicates white. The picture he is envisioning is also N millimeters long and the ith millimeter consists purely of the color Ci. Unfortunately, his brush isn't fine enough to paint every millimeter one by one. The brush is 3 millimeters wide and so it can only paint three millimeters at a time with the same color. Painting over the same place completely replaces the color by the new one. Also, Chef has lots of bottles of paints of each color, so he will never run out of paint of any color.
Chef also doesn't want to ruin the edges of the canvas, so he doesn't want to paint any part beyond the painting. This means, for example, Chef cannot paint just the first millimeter of the canvas, or just the last two millimeters, etc.
Help Chef by telling him whether he can finish the painting or not with these restrictions.

-----Input-----
The first line of the input contains an 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 C1, C2, ..., CN denoting the colors of Chef's painting.

-----Output-----
For each test case, output a single line containing either “Yes” or “No” (without quotes), denoting whether Chef can finish the painting or not.

-----Constraints-----
- 1 ≤ T ≤ 105
- 3 ≤ N ≤ 105
- The sum of the Ns over all the test cases in a single test file is ≤ 5×105
- 1 ≤ Ci ≤ 105

-----Example-----
Input:3
4
1 5 5 5
4
1 1 1 5
3
5 5 2
Output:Yes
Yes
No

-----Explanation-----
Example case 1. Chef's canvas initially contains the colors [0,0,0,0]. Chef can finish the painting by first painting the first three millimeters with color 1, so the colors become [1,1,1,0], and then the last three millimeters with color 5 so that it becomes [1,5,5,5].
Example case 2. Chef's canvas initially contains the colors [0,0,0,0]. Chef can finish the painting by first painting the last three millimeters by color 5 so the colors become [0,5,5,5], and then the first three millimeters by color 1 so it becomes [1,1,1,5].
Example case 3. In this test case, Chef can only paint the painting as a whole, so all parts must have the same color, and the task is impossible.
user: Kefaa has developed a novel decomposition of a tree. He claims that this decomposition solves many difficult problems related to trees. However, he doesn't know how to find it quickly, so he asks you to help him.
You are given a tree with $N$ vertices numbered $1$ through $N$. Let's denote an edge between vertices $u$ and $v$ by $(u, v)$. The triple-tree decomposition is a partition of edges of the tree into unordered triples of edges $(a, b), (a, c), (a, d)$ such that $a \neq b \neq c \neq d$. Each edge must belong to exactly one triple.
Help Kefaa with this problem — find a triple-tree decomposition of the given tree or determine that no such decomposition exists.

-----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 $N$.
- Each of the following $N-1$ lines contains two space-separated integers $u$ and $v$ describing an edge between vertices $u$ and $v$ of the tree.

-----Output-----
- For each test case, print a line containing the string "YES" if a triple-tree decomposition of the given tree exists or "NO" otherwise.
- If it exists, print $\frac{N-1}{3}$ more lines describing a decomposition.
- Each of these lines should contain four space-separated integers $a$, $b$, $c$ and $d$ describing a triple of edges $(a, b), (a, c), (a, d)$.
If more than one triple-tree decomposition exists, you may output any one.

-----Constraints-----
- $1 \le T \le 100$
- $2 \le N \le 2 \cdot 10^5$
- $1 \le u, v \le N$
- the sum of $N$ over all test cases does not exceed $2 \cdot 10^5$

-----Subtasks-----
Subtask #1 (20 points): $2 \le N \le 10$
Subtask #2 (30 points):$2 \le N \le 5000$ and the sum of $N$ overall testcases doesn't exceed $5000$
Subtask #3 (50 points): original constraints

-----Example Input-----
2
4
1 2
1 3
1 4
7
1 2
2 3
1 4
4 5
1 6
6 7

-----Example Output-----
YES
1 2 3 4
NO
user: Sandu, a teacher in Chefland introduced his students to a new sequence i.e.
0,1,0,1,2,0,1,2,3,0,1,2,3,4........
The Sequence starts from 0 and increases by one till $i$(initially i equals to 1), then repeat itself with $i$ changed to $i+1$
Students being curious about the sequence asks the Nth element of the sequence. Help Sandu to answer the Students

-----Input:-----
- The first-line will contain $T$, the number of test cases. Then the test case follows. 
- Each test case contains a single numbers N.

-----Output:-----
Print the Nth element of the sequence

-----Constraints-----
- $1 \leq T \leq 1000$
- $1 \leq N  \leq 10^{18}$

-----Sample Input:-----
5
8
9
20
32
109

-----Sample Output:-----
2
3
5
4
4
user: Ayu loves distinct letter sequences ,a distinct letter sequence is defined by a sequence of small case english alphabets such that no character appears more then once.
But however there are two phrases that she doesn't like these phrases are "kar" and "shi" and she is given a sequence of distinct characters and she wonders how many such sequences she can form using all the characters such that these phrases don't occur.
Help her finding the number of such sequences.
New Year Gift - It is guaranteed that for sequences of length greater then 6 letters k,a,r,s,h,i will be present(we thought of being generous, thank us later :)).

-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow. 
- Each line consists of a string $S$ (3<=s.length<=18) of distinct characters.

-----Output:-----
Print the number of sequences that can be formed by permuting all the characters such that phrases "kar" and "shi" don't occur.

-----Constraints-----
- $1 \leq T \leq 10$
- $3 \leq S.length \leq 18$

-----Sample Input:-----
2
karp
abcd

-----Sample Output:-----
22
24