meetforever/GRPO_CL huggingface dataset:
user: Note : This question carries $100$ $points$
CodeLand is celebrating a festival by baking cakes! In order to avoid wastage, families follow a unique way of distributing cakes.
For $T$ families in the locality, $i$-th family (1 <= $i$ <= $T$) has $N$ members. They baked $S$ slices of cakes. The smallest member of the family gets $K$ slices of cakes. Each family has a lucky number of $R$ and they agree to distribute the slices such that the member gets $R$ times more slices than the member just smaller than them. Since the family is busy in festival preparations, find out if the number of slices would be sufficient for the family or not. Also, find how many extra slices they have or how many slices are they short of.
Also, the locality is kind and believes in sharing. So, you also need to determine if each family would have sufficient slices if families shared their cakes among each other!

-----Input :-----
- First line of input will have a single integer $T$ i.e. the number of families in the locality
- For next $T$ lines, each line will describe one family through 4 integers i.e. $S$, $N$, $K$, $R$ separated by spaces

-----Output-----
- First $T$ lines of output will show if slices are enough for the family or not, followed by extra or required slices. For example, if slices are sufficient, the output would be $POSSIBLE$ $10$ which implies 10 slices are extra. If slices are insufficient, the output would be $IMPOSSIBLE$ $15$ which implies 15 slices are required to fulfill the requirement.
- Last line of output would be $IMPOSSIBLE$ or $POSSIBLE$ depending on whether each family would get enough slices after sharing.

-----Constraints:-----
- 1 ≤ $T$ ≤ 50
- 1 ≤ $N$ ≤ 15
- 1 ≤ $S$ ≤ 10^9
- 1 ≤ $K$ ≤ 5
- 1 ≤ $R$ ≤ 5

-----Sample Input:-----
5
100 4 2 2
100 4 3 2
100 4 3 3
200 4 4 2
10 3 2 2

-----Sample Output:-----
POSSIBLE 70
POSSIBLE 55
IMPOSSIBLE 20
POSSIBLE 140
IMPOSSIBLE 4
POSSIBLE

-----Explanation-----
For the first home, the youngest member has got 2 slices, the one older than them gets 2 times more slices, and so forth. So, first home needs 2 + 4 + 8 + 16 = 30 slices. But since they made 100, they have 70 extra, hence we print 'POSSIBLE 70'.
Doing the same for other houses, we find 20 + 4 = 24 slices extra are required so that all families have enough slices. In this case we have 70 + 55 + 140 = 265 extra slices which is > 24, so we print 'POSSIBLE' in last line.
user: You are given a tree rooted at node $1$ with $N$ vertices. The $i$$th$ vertex initially has value $A_i (1 \leq i \leq N)$. You are also given $Q$ queries. 
In each query you are given a vertex $V$.  Let $S = \{ S_1 , S_2 , ... S_x \} $ denote the set of vertices such that $S_i$ is in the subtree of $V$, distance between $S_i$ and $V$ is even and $S_i \neq V$ for all $i$. For all $S_i$ , add $A$$S_i$ to $A_V$ and change the value of $A$$S_i$ to zero.
Find the values of all the vertices after all queries are performed. 
Note-The distance between two vertices is defined as the number of edges
traversed on the shortest path from one vertex to the other.

-----Input:-----
- The first line contains an integer $T$ denoting the number of test cases.
- The first line of each test case contain two integers $N$ and $Q$.
- The second line contains $N$ space separated integers, $A_1, A_2, ..., A_n$ denoting the initial values of the vertices.
- The next $N-1$ lines contain two integers $u$ and $v$ denoting an edge between $u$and $v$.
- The next $Q$ lines contain a single integer which is the query.

-----Output:-----
- Print a single line containing $N$ integers for each test case which is the final values of the vertices.

-----Constraints:-----
- $1\leq T \leq 10$
- $1 \leq N \leq 200000$
- $1 \leq Q \leq 200000$
- $0 \leq A_i \leq 10^9$
- The sum of $N$ over all test cases does not exceed $200000$.
- The sum of $Q$ over all test cases does not exceed $200000$.

-----Sample Input-----
1
4 3
6 2 7 3
1 2
2 3
3 4
3
2
1

-----Sample Output-----
13 5 0 0

-----Explanation-----
Node $3$ has no child in its subtree which is at  an even distance so there is no change in the values. Values of nodes after $1st$ query: $6, 2, 7, 3$.
Node $4$ is at an even distance in the subtree of node $2$ so $A_4$ gets added to $A_2$ and $A_4$ becomes 0. Values of nodes after $2nd$ query: $6, 5, 7, 0$.
Node $3$ is at an even distance in the subtree of node $1$ so $A_3$ gets added to $A_1$ and $A_3$ becomes 0. Values of nodes after $3rd$ query: $13, 5, 0, 0$.
user: This is probably the simplest problem ever. You just need to count the number of ordered triples of different numbers (X1, X2, X3), where Xi could be any positive integer from 1 to Ni, inclusive (i = 1, 2, 3).
No, wait. I forgot to mention that numbers N1, N2, N3 could be up to 1018. Well, in any case it is still quite simple :)
By the way, because of this the answer could be quite large. Hence you should output it modulo 109 + 7. That is you need to find the remainder of the division of the number of required triples by 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. The only line of each test case contains three space-separated integers N1, N2, N3.

-----Output-----
For each test case, output a single line containing the number of required triples modulo 109 + 7.

-----Constraints-----
- 1 ≤ T ≤ 1000
- 1 ≤ Ni ≤ 1018

-----Example-----
Input:
5
3 3 3
2 4 2
1 2 3
25 12 2012
1 1 2013

Output:
6
4
1
578880
0

-----Explanation-----
Example case 1. We have the following triples composed of different numbers up to 3:
(1, 2, 3)
(1, 3, 2)
(2, 1, 3)
(2, 3, 1)
(3, 1, 2)
(3, 2, 1)
Example case 2. Here the triples are:
(1, 3, 2)
(1, 4, 2)
(2, 3, 1)
(2, 4, 1)
Example case 3. Here the only triple is (1, 2, 3).
Example case 4. Merry Christmas! 
Example case 5. ... and Happy New Year! By the way here the answer is zero since the only choice for X1 and for is X2 is 1, so any such triple will have equal numbers.
user: Chef recently learned about concept of periodicity of strings. A string is said to have a period P, if P divides N and for each i, the i-th of character of the string is same as i-Pth character (provided it exists), e.g. "abab" has a period P = 2, It also has a period of P = 4, but it doesn't have a period of 1 or 3.
Chef wants to construct a string of length N that is a palindrome and has a period P. It's guaranteed that N is divisible by P. This string can only contain character 'a' or 'b'. Chef doesn't like the strings that contain all a's or all b's.
Given the values of N, P, can you construct one such palindromic string that Chef likes? If it's impossible to do so, output "impossible" (without quotes)

-----Input-----
The first line of the input contains an integer T denoting the number of test cases.
The only line of each test case contains two space separated integers N, P.

-----Output-----
For each test case, output a single line containing the answer of the problem, i.e. the valid string if it exists otherwise "impossible"  (without quotes). If there are more than possible answers, you can output any.

-----Constraints-----
- 1 ≤ T ≤ 20
- 1 ≤ P, N ≤ 105

-----Subtasks-----
- Subtask #1 (25 points) : P = N
- Subtask #2 (75 points) : No additional constraints

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

Output
impossible
impossible
aba
abba
abaaba

-----Explanation-----
Example 1: The only strings possible are either aaa or bbb, which Chef doesn't like. So, the answer is impossible.
Example 2: There are four possible strings, aa, ab, ba, bb. Only aa and bb are palindromic, but Chef doesn't like these strings. Hence, the answer is impossible.
Example 4: The string abba is a palindrome and has a period of 4.
Example 5: The string abaaba is a palindrome and has a period of length 3.
user: Chef has some numbers. His girlfriend Chefina feels good when chef gives her a particular pattern number also called as Armstrong number.
Armstrong number is a number whose sum of its all individual digit raise to the power of the number of digit in that number is equal to that number itself
eg.. 153 = 1^3 + 5^3 + 3^3 (153 is an Armstrong number)
1634 = 1^4 + 6^4 + 3^4 + 4^4 (1634 is an Armstrong number)
As a love guru of chef you have to help chef to find Armstrong numbers Among the numbers which chef has initially so that Chefina feels good

-----Input:-----
First line will contain a positive Integer $T$ which is the number of testcases
Next $T$ lines follows an Integer $N$.

-----Output:-----
For Every n You have to print "FEELS GOOD" without qoutes if it is an armstrong number otherwise Print "FEELS BAD" without quotes

-----Constraints-----
- $1 \leq T \leq 10$
- $2 \leq N \leq 10^6$

-----Sample Input:-----
3
153
11
1634

-----Sample Output:-----
FEELS GOOD
FEELS BAD
FEELS GOOD

-----EXPLANATION:-----
For test case 1 --> 153 = 1^3 + 5^3 + 3^3 (153 is an armstrong number)
user: After a long and successful day of preparing food for the banquet, it is time to clean up. There is a list of n jobs to do before the kitchen can be closed for the night. These jobs are indexed from 1 to n.

Most of the cooks have already left and only the Chef and his assistant are left to clean up. Thankfully, some of the cooks took care of some of the jobs before they left so only a subset of the n jobs remain. The Chef and his assistant divide up the remaining jobs in the following manner. The Chef takes the unfinished job with least index, the assistant takes the unfinished job with the second least index, the Chef takes the unfinished job with the third least index, etc. That is, if the unfinished jobs were listed in increasing order of their index then the Chef would take every other one starting with the first job in the list and the assistant would take every other one starting with the second job on in the list.

The cooks logged which jobs they finished before they left. Unfortunately, these jobs were not recorded in any particular order. Given an unsorted list
of finished jobs, you are to determine which jobs the Chef must complete and which jobs his assitant must complete before closing the kitchen for the 
evening.

-----Input-----

The first line contains a single integer T ≤ 50 indicating the number of test cases to follow. Each test case consists of two lines. The first line contains two numbers n,m satisfying 0 ≤ m ≤ n ≤ 1000. Here, n is the total number of jobs that must be completed before closing and m is the number of jobs that have already been completed. The second line contains a list of m distinct integers between 1 and n. These are the indices of the jobs that have already been completed. Consecutive integers are separated by a single space.

-----Output-----

The output for each test case consists of two lines. The first line is a list of the indices of the jobs assigned to the Chef. The second line is a list of the indices of the jobs assigned to his assistant. Both lists must appear in increasing order of indices and consecutive integers should be separated by a single space. If either the Chef or the assistant is not assigned any jobs, then their corresponding line should be blank.

-----Example-----
Input:
3
6 3
2 4 1
3 2
3 2
8 2
3 8

Output:
3 6
5
1

1 4 6
2 5 7
user: Let's consider a rooted binary tree with the following properties:
- The number of nodes and edges in the tree is infinite
- The tree root is labeled by $1$
- A node labeled by $v$ has two children: $2 \cdot v$ (the left child of $v$), and $2 \cdot v + 1$ (the right child of $v$).
Here is an image of the first several layers of such a tree:

Let's consider four operations that you are allowed to apply during the tree traversal:
- move to the left child - move from $v$ to $2 \cdot v$
- move to the right child - move from $v$ to $2 \cdot v + 1$
- move to the parent as a left child - move from $v$ to $\frac{v}{2}$ if $v$ is an even integer
- move to the parent as a right child - move from $v$ to $\frac{v - 1}{2}$ if $v$ is an odd integer
It can be proven, that for any pair of nodes $u$ and $v$, there is only one sequence of commands that moves from $u$ to $v$ and visits each node of the tree at most once. Let's call such a sequence of commands a path configuration for a pair of nodes $(u, v)$.
You are asked to process a series of the following queries: 
You are given three integers $n$, $u$ and $v$ ($1 \leq u, v \leq n$). Count the pairs of nodes $(w, t)$ ($1 \leq w, t \leq n$) such that the path configuration for $(w, t)$ is the same with the path configuration for $(u, v)$.

-----Input-----
- The first line of input contains a single integer $Q$, denoting the number of queries to process.
- Each of the next $Q$ lines contains three space-separated integers $n$, $u$ and $v$ denoting a query.

-----Output-----
For each query, print the answer on a separate line.

-----Constraints-----
- $1 \leq Q \leq 2 \cdot 10^4$
- $1 \leq u, v \leq n \leq 10^{9}$

-----Example Input-----
3
11 9 11
10 2 2
8 1 8

-----Example Output-----
2
10
1

-----Explanation-----
In the first query from the example test case, you should count pairs $(5, 7)$ and $(9, 11)$.
In the second query from the example test case, you should count the following pairs: $(1, 1)$, $(2, 2)$, $(3, 3)$, $(4, 4)$, $(5, 5)$, $(6, 6)$, $(7, 7)$, $(8, 8)$, $(9, 9)$ and $(10, 10)$.
In the third query from the example test case, you should only count a pair $(1, 8)$.
user: Chef taught his brother Chefu about right angled triangle and its properties. Chefu says that he has understood everything about right angled triangles. Chef wants to check learning of his brother by asking the following question "Can you find a right angled triangle whose length of hypotenuse  is H and its area is S?"
Chefu is confused how to solve it. I hope you are not. Please solve this by finding a right angled triangle with hypotenuse H and area S. If it not possible to do so, then output -1.

-----Input-----
The first line of the input contains a single integer T denoting the number of test-cases. T test cases follow.
For each test case, there will be a single line containing two space separated integers H and S.

-----Output-----
Output the answer for each test-case in a single line. If it is not possible to find such a triangle, output -1. Otherwise print 3 real numbers corresponding to the lengths of the sides of the triangle sorted in non-decreasing order. Please note that the length of the triangle sides should not differ by more than 0.01 in absolute value from the correct lengths.

-----Constraints-----
- 1 ≤ T ≤ 105
- 1 ≤ H ≤ 106
- 1 ≤ S ≤ 1012

-----Example-----
Input:4
5 6
6 10
258303 89837245228
616153 77878145466
Output:3.00000 4.00000 5.00000
-1
-1
285168.817674 546189.769984 616153.000000
user: Help Saurabh with his Chemistry Assignment.

Saurabh has been given a chemistry assignment by Ruby Mam. Though the assignment is simple but

Saurabh has to watch India vs Pakistan Match and he has no time to do the assignment by himself.

So Saurabh wants you to do his assignment so that he doesn’t get scolded by Ruby Mam . The assignment

is as follows , Suppose there are X particles initially at time t=0 in a box. At a time t the number of particles in

box becomes t times the number of particles at time t-1 . You will be given N and X where N is time at which the

number of particles in box is to be calculated and X is the number of particles at time t=0.

-----Input-----
The first line will contain the integer T, the number of test cases. Each test case consists of two space

separated integers N and X .

-----Output-----
For each test case, output the answer to the query. Since the output can be very large, output the answer modulo

10^6+3

-----Constraints-----
- 1 ≤ T ≤ 100000
- 1 ≤ N,X ≤ 10^18

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

Output:
2
2

-----Explanation-----
Example case 2.At t=0 particles are 1 ,so at t=1 ,particles are 1*1 = 1 particles. At t=2, particles are 2*1 = 2 particles.
user: Given the values at the leaf nodes of a complete binary tree. The total number of nodes in the binary tree, is also given. Sum of the values at both the children of a node is equal to the value of the node itself. You can add any value or subtract any value from a node. Print the minimum change(difference made) in the sum of values of all the nodes in the tree, such that all the leaf nodes have the same value.

Note: If a value transfers from one node to another, then that is not a change, but if an extra is needed to be added or subtracted to the entire total value of the nodes, then that is a change.

Input Description:

Input will contain an integer N, the number of nodes in the tree on a newline, followed by N space separated integers representing the values at the leaf nodes of the tree.

Output Description:

Print the required value on a newline.

Constraints:

1<=N<=20000
1<=Value at each node in the leaves<=1000

Example 1:
Input:

1

50
Output:

0

Explanation: Since there is only one node, it is a leaf node itself and no change needs to be made.

Example 2:
Input:

3

200 800
Output:

0 

Explanation: There are two leaf nodes, and they can be made to 500 500, since no change in the total was made so difference made is 0. 

Example 3:
Input:

30

29 33 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Output:

6

Output: A total change of 6 needs to be changed to the entire value of the nodes, to get the leaf nodes equal.
user: A manufacturing project consists of exactly $K$ tasks. The board overviewing the project wants to hire $K$ teams of workers — one for each task. All teams begin working simultaneously.
Obviously, there must be at least one person in each team. For a team of $A$ workers, it takes exactly $A$ days to complete the task they are hired for. Each team acts independently, unaware of the status of other teams (whether they have completed their tasks or not), and submits their result for approval on the $A$-th day.
However, the board approves the project only if all $K$ teams complete their tasks on the same day — it rejects everything submitted on any other day. The day after a team finds out that its result was rejected, it resumes work on the same task afresh. Therefore, as long as a team of $A$ workers keeps getting rejected, it submits a new result of their task for approval on the $A$-th, $2A$-th, $3A$-th day etc.
The board wants to hire workers in such a way that it takes exactly $X$ days to complete the project. Find the smallest number of workers it needs to hire.

-----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 $K$ and $X$.

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

-----Constraints-----
- $1 \le T \le 40$
- $2 \le K, X \le 10^6$

-----Example Input-----
2
2 3
2 6

-----Example Output-----
4
5

-----Explanation-----
Example case 1: We can hire a team of $3$ workers for task $1$ and $1$ worker for task $2$. The one-man team working on task $2$ completes it and submits the result for approval on each day, but it is rejected on the first and second day. On the third day, the team working on task $1$ also completes their task, so the project gets approved after exactly $3$ days.
Example case 2: We can hire a team of $3$ workers for task $1$ and a team of $2$ workers for task $2$.
user: Consider a 2d-grid. That is, each cell is identified by (i,j). You have received reports of two snake-sightings on this grid. You want to check whether they could be partial sightings of the same snake or not.
Each of the snake sightings correspond to a straight, axis-parallel line segment in the grid, and the starting and ending cells for each are given to you. Now consider a graph, where each cell in the 2d-grid is a vertex. And there is an edge between 2 vertices if and only if the cells corresponding to these two vertices are consecutive cells in at least one of the two snakes. That is, at least in one of the snakes, when you go from one end point to the other end point, these two cells should occur consecutively.
The two sightings/snakes are said to be same, if both these conditions are satisfied:
- The union of the set of cells in the first snake and the set of cells in the second snake, should form a connected component in this graph.
- No vertex should have degree more than 2 in the graph.
In other words, the induced subgraph on the union set must be a path graph.

-----Input-----
- The first line contains a single integer, T, which is the number of testcases. The description of each testcase follows.
- The first line of each testcase contains four integers: X11, Y11, X12, Y12. This represents the fact that the first snake's end points are (X11, Y11) and (X12, Y12).
- The second line of each testcase contains four integers: X21, Y21, X22, Y22. This represents the fact that the second snake's end points are (X21, Y21) and (X22, Y22).

-----Output-----
- For each testcase, output "yes" if the snakes are the same, as per the definition given above. Output "no" otherwise.

-----Constraints-----
- 1 ≤ T ≤ 105
- -109 ≤ Xij,Yij ≤ 109
- The two end points of every snake is guaranteed to be either on the same row or on the same column. Thus, the snake occupies all the cells between these cells, including the end points.

-----Example-----
Input:
4
2 1 8 1
11 1 7 1
2 1 8 1
11 1 9 1
2 1 8 1
3 1 3 -2
2 1 8 1
2 1 2 -2
Output:
yes
no
no
yes

-----Explanation-----
In the images, the first snake is red, the second snake is yellow, and the intersections, if any, are in orange.
The first test case corresponds to:

Both the conditions on the graph are satisfied, and hence this is a "yes".
The second test case corresponds to:

There is no edge between the vertex corresponding to the (8,1) cell and the vertex corresponding to (9,1), Hence, the union set is disconnected, and thus the answer is "no". 
The third test case corresponds to:

The vertex corresponding to the cell (3,1) has degree 3, which is more than 2, and thus the answer is "no". 
The fourth test case corresponds to:

Both the conditions on the graph are satisfied, and hence this is a "yes".
user: Ada is playing pawn chess with Suzumo.
Pawn chess is played on a long board with N$N$ squares in one row. Initially, some of the squares contain pawns.
Note that the colours of the squares and pawns do not matter in this game, but otherwise, the standard chess rules apply:
- no two pawns can occupy the same square at the same time
- a pawn cannot jump over another pawn (they are no knights!), i.e. if there is a pawn at square i$i$, then it can only be moved to square i−2$i-2$ if squares i−1$i-1$ and i−2$i-2$ are empty
- pawns cannot move outside of the board (outs are forbidden)
The players alternate turns; as usual, Ada plays first. In each turn, the current player must choose a pawn and move it either one or two squares to the left of its current position. The player that cannot make a move loses.
Can Ada always beat Suzumo? Remember that Ada is a chess grandmaster, so she always plays optimally.

-----Input-----
- The first line of the input contains a single integer T$T$ denoting the number of test cases. The description of T$T$ test cases follows.
- The first and only line of each test case contains a single string S$S$ with length N$N$ describing the initial board from left to right. An empty square and a square containing a pawn are denoted by the characters '.' and 'P' respectively.

-----Output-----
For each test case, print a single line containing the string "Yes" if Ada wins the game or "No" otherwise (without quotes).

-----Constraints-----
- 1≤T≤500$1 \le T \le 500$
- 2≤N≤128$2 \le N \le 128$
- S$S$ contains only characters '.' and 'P'

-----Example Input-----
1
..P.P

-----Example Output-----
Yes

-----Explanation-----
Example case 1: Ada can move the first pawn two squares to the left; the board after this move looks like
P...P

and now, Suzumo can only move the second pawn. If he moves it one square to the left, Ada will move it two squares to the left on her next move, and if he moves it two squares to the left, Ada will move it one square to the left, so the board after Ada's next move will look like
PP...

and Suzumo cannot make any move here.