meetforever/GRPO_CL huggingface dataset:
user: Baxter Scott owns The Enlightened Dairy Co., a dairy company with magical cows. Early each morning, he brushes his teeth, strolls outside, and finds that the cows have doubled in number. With double the number of cows, he can produce double the quantity of milk. While he is ecstatic that he has so many cows and so much milk, the Dairy Regulator forces him to keep at most $C$ cows on any given farm, which greatly complicates his business.

At The Enlightened Dairy Co., Baxter has access to an unlimited number of farms, each with a maximum capacity of $C$ cows. On each farm, cows reproduce at the same rate: they always double in number when the clock strikes midnight. To stay within the Regulator’s rules, whenever a farm has strictly more than $C$ cows, Baxter selects half of the cows on that farm and moves them to an entirely new, empty farm. More precisely, if there are $D \leq C$ cows on a farm, he leaves all $D$ cows on the farm, but if there are $D > C$ cows on a farm, he leaves $\left\lceil \frac{D}{2} \right\rceil $ cows on the farm and takes $\left\lfloor \frac{D}{2} \right\rfloor $ cows to a new, empty farm. (Here $\left\lceil \ \right\rceil $ and $\left\lfloor \ \right\rfloor $ denote the ceiling and floor functions, which round up/down to the nearest integer, respectively.) He does this early every morning, before the Regulator could possibly show up, so that he can avoid paying hefty Moo Fees.

The Regulator needs to know how many farms she will be inspecting when she visits The Enlightened Dairy Co. The Regulator inspects every farm that has at least one cow, and does not inspect any farm with zero cows. Given the number of cows on each farm with at least one cow on Day $0$, compute the number of farms that need inspecting on any given day.

-----Input-----
The first line of input contains three space-separated integers, $C$, $N$, and $M$, where $C$ ($1 \leq C \leq 1000$) is the maximum number of cows allowed on a farm, $N$ ($0 \leq N \leq 1000$) is the number of farms with at least one cow on Day $0$, and $M$ ($1 \leq M \leq 50$) is the number of different days on which the Regulator visits. The following $N$ lines of input each have a single integer $c_ i$($1 \leq c_ i \leq C$), representing the number of cows on the $i$th of the $N$ farms on Day $0$. The final $M$ lines each have a single integer $d_ j$ ($0 \leq d_ j \leq 50$), representing a day when the Regulator visits. All $d_ j$ are distinct.

-----Output-----
For each $d_ j$, output one line with a single integer: the number of farms the Regulator will inspect on Day $d_ j$.

-----Examples-----
Sample Input:
1 5 5
1
1
1
1
1
0
1
2
3
4
Sample Output:
5
10
20
40
80
user: A Fenwick Tree (also known as a Binary Indexed Tree) is a data structure on an array which enables fast ($O(\log n)$) updates and prefix sum queries on the underlying data.

For this problem, implement a Fenwick Tree to support operations of two types: (a) increment an element in the array or (b) query the prefix sum of a portion of the array.

-----Input-----
The first line of input contains two integers $N$, $Q$, where $1 \le N \le 5000000$ is the length of the array and $0 \le Q \le 5000000$ is the number of operations. Then follow $Q$ lines giving the operations. There are two types of operations:
 - “+ $i$ $\delta $” indicates that $a[i]$ is incremented by $\delta $, where $0 \le i < N$ and $-10^9 \le \delta \le 10^9$ (both are integers)
 - “? $i$” is a query for the value of $a[0] + a[1] + \ldots + a[i-1]$, where $0 \le i \le N$ (for $i = 0$ this is interpreted as an empty sum)

You should assume that every array entry is initially $0$.

-----Output-----
For each query in the input, output one line giving the answer to that query.

-----Examples-----
Sample Input:
10 4
+ 7 23
? 8
+ 3 17
? 8
Sample Output:
23
40
user: At some point or another, most computer science students have written a standard Sudoku solving program. This is yet another “put numbers in a grid” puzzle.

Numbers are placed in the grid so that each outlined region contains the numbers $1$ to $n$, where $n$ is the number of squares in the region. The same number can never touch itself, not even diagonally.



Incomplete Grid

Solution Grid

For this problem, you will write a program that takes as input an incomplete puzzle grid and outputs the puzzle solution grid.

-----Input-----
The input consists of a single data set. This data set starts with a line containing the number of rows in the input grid $R$, ($1 \le R \le 7$), and the number of columns in the input grid $C$, ($1 \le C \le 7$), separated by spaces. The next $R$ lines contain a representation of the incomplete input grid, one row per line. Each row has $C$ characters, representing the digits in each cell. The value in each cell is represented by either the digit already in that cell or a ‘-’ for an initially empty cell.

This grid is followed by a description of the separate regions in the grid. The first of these lines specifies the total number of regions. This is followed by one line for each region that specifies the cells contained in that region. Each region description consists of a decimal number $N$, specifying the number of cells in the region, followed by $N$ cell descriptions separated by spaces. Each cell description consists of a left parenthesis, followed the cell’s row index, followed by a comma, followed by the cell’s row number, followed by a right parenthesis. Regions do not overlap.

-----Output-----
Output $R$ lines containing $C$ digits (separated by single spaces) showing the solution grid for the corresponding input data set. You are guaranteed that there is a unique solution.

-----Examples-----
Sample Input:
3 5
- - - - -
- - - - -
4 - - - 1
5
1 (1,1)
2 (1,2) (1,3)
5 (2,1) (2,2) (3,1) (3,2) (3,3)
4 (2,3) (2,4) (1,4) (1,5)
3 (3,4) (3,5) (2,5)
Sample Output:
1 2 1 2 1
3 5 3 4 3
4 2 1 2 1
user: “It’s like how hot dogs come in packs of ten, and buns come in packs of eight or twelve — you have to buy nine packs to make it come out even.” 

This is a quote from the 1986 movie, “True Stories”, and it’s true; well, almost true. You could buy four packs of $10$ hotdogs and five packs of $8$ buns. That would give you exactly $40$ of each. However, you can make things even with fewer packs if you buy two packs of $10$ hotdogs, along with a pack of $8$ buns and another pack of $12$ buns. That would give you $20$ of each, using only $4$ total packs.

For this problem, you’ll determine the fewest packs you need to buy to make hotdogs and buns come out even, given a selection of different bun and hotdog packs available for purchase.

-----Input-----
The first input line starts with an integer, $H$, the number of hotdog packs available. This is followed by $H$ integers, $h_1 \ldots h_ H$, the number of hotdogs in each pack. The second input line starts with an integer, $B$, giving the number of bun packs available. This is followed by $B$ integers, $b_1 \ldots b_ B$, indicating the number of buns in each pack. The values $H$ and $B$ are between $0$ and $100$, inclusive, and the sizes of the packs are between $1$ and $1000$, inclusive. Every available pack is listed individually. For example, if there were five eight-bun packs available for purchase, the list of bun packs would contain five copies of the number eight.

-----Output-----
If it’s not possible to purchase an equal number of one or more hotdogs and buns, just output “impossible”. Otherwise, output the smallest number of total packs you can buy (counting both hotdog and bun packs) to get exactly the same number of hotdogs and buns.

-----Examples-----
Sample Input:
4 10 10 10 10
10 8 8 8 12 12 12 8 8 12 12
Sample Output:
4
user: Marek and his schoolmates have just finished their studies at the university. They wanted to celebrate it with a game of paintball. After an hour of playing a very strange thing happened – everyone had exactly one bullet left. Marek, being a very curious person, wanted to know whether it’s possible that everyone will be hit exactly once provided nobody moves.

-----Task-----
You are given a description of the situation during a paintball game when every player has only one bullet. The description of the game consists of pairs of players who can see each other. If a player can see another player, he can fire at him. Your task is to find a target for each player such that everyone will be hit.

-----Input-----
The first line of input contains two space separated integers $N$ and $M$, satisfying $2\leq N\leq 1000$ and $0\leq M\leq 5000$, where $N$ is the number of players. Players are numbered $1, 2, \ldots , N$. $M$ lines follow, each line containing two space separated integers $A$ and $B$ ($1\leq A < B\leq N$), denoting that players $A$ and $B$ can see each other. Each pair of players appears at most once in the input.

-----Output-----
If there is no assignment of targets such that everyone will be hit, output Impossible. Otherwise output $N$ lines. The $i$-th line should contain the number of the target of the $i$-th player. If there is more than one solution, output any one.

-----Examples-----
Sample Input:
3 3
1 2
2 3
1 3
Sample Output:
2
3
1
user: Boris works at a secret communication station for the government training new employees on how to encode messages. Messages at this station are encoded with a rotation (“Caesar”) cipher that replaces each letter with one at a set offset in the alphabet, e.g., for offset $2$ an ‘a’ is replaced by a ‘c’ and ‘y’ is replaced by ‘a’. In order to encode a message, this cipher may be applied more than once. Each such application counts as a step in the encoding process. Boris teaches the new employees to encode messages by demonstrating each step of the encoding individually. However, Boris does not does not like seeing messages that contain at least half as many vowels as consonants after applying an encoding step. (Boris considers ‘a’, ‘e’, ‘i’, ‘o’, ‘u’, and ‘y’ as vowels.) He grows annoyed and more and more unhappy with each step where this situation occurs.

Your job is to encode a given message and determine whether Boris will see fewer steps that annoy him than ones that don’t. Since Boris wants to be happy in his job he will give a message that annoys him too much to a colleague.

-----Input-----
The first line of the input contains a single integer $O$ ($1 \le O \le 25$) that represent the offset used for the rotation cipher. The second line contains the message to encode, which consists entirely of lowercase English characters. The length of the message is between $1$ and $80$ characters. The third line contains an integer $N$ ($1 \le N \le 26$), the number of times the cipher must be applied.

-----Output-----
Output ‘Boris’ if strictly more steps sound good than bad, and ‘Colleague’ otherwise.

-----Examples-----
Sample Input:
1
thequickbrownfoxjumpedoverthelazydog
10
Sample Output:
Boris
user: Simon is a contestant on the new exciting game show, Add ’Em Up! In this show, contestants are given a set of cards with multi-digit numbers on them, and need to select two cards that add up to a given sum. However, to make things more interesting, the font that the cards are printed in (see below) means that certain cards will display different numbers if turned upside-down, and it is legal to turn either or both of the cards if necessary to generate the sum. Please help Simon win the game!

-----Input-----
The first line contains two integers, $n$, the number of cards, and $s$, the desired sum. The second line will contain $n$ integers between $1$ and $100000000$ inclusive. You may assume that $1\leq n\leq 100000$ and $2\leq s\leq 200000000$.

-----Output-----
The output should be a single line consisting of the string YES if two cards can be chosen such that (in some orientation) they add up to $s$, and the string NO otherwise.

-----Examples-----
Sample Input:
3 66
15 21 22
Sample Output:
NO
user: Michael works in a bakery. At the end of his shift, his boss wants the breads sorted in a certain order. She can’t seem to decide on which order, though – every day there seems to be a new one – to Michael’s despair. Michael has worked there for a while now and has learned a neat trick with his wooden bakery paddle. He can take three breads next to each other on his paddle and throw them up in the air such that when they land, the right-most has moved to the left-most position, and the two other breads have moved one place to the right. In other words, he can rotate to the right a subsequence of breads of length three.

Before the end of the shift, his coworkers place the breads in a long line. Michael would like to sort the line of breads using his paddle trick. He can take any three consecutive breads along the line on his paddle, rotate them, and then put them back. Sometimes though, it does not matter how many times he uses his paddle – the line of bread just doesn’t seem to be possible to sort the way the boss wants…

-----Input-----
The first line of input contains a positive integer $N, (3 \leq N \leq 100000)$, denoting the number of breads. Then two lines follow: On the first line, a permutation of the integers $1$ through $N$ describing the order in which the breads are lined up. On the second line, a permutation of the integers $1$ through $N$ describing how Michael’s boss wants the breads sorted.

-----Output-----
Output “Possible” if Michael can sort the breads with his paddle in the order prescribed by his boss, otherwise “Impossible”.

-----Examples-----
Sample Input:
4
1 3 4 2
4 3 2 1
Sample Output:
Possible
user: Iskander the Baker is decorating a huge cake, covering the rectangular surface of the cake with frosting. For this purpose, he mixes frosting sugar with lemon juice and food coloring, in order to produce three kinds of frosting: yellow, pink, and white. These colors are identified by the numbers $0$ for yellow, $1$ for pink, and $2$ for white.

To obtain a nice pattern, he partitions the cake surface into vertical stripes of width $A_1, A_2, \dots , A_ n$ centimeters, and horizontal stripes of height $B_1, B_2, \dots , B_ n$ centimeters, for some positive integer $n$. These stripes split the cake surface into $n\times n$ rectangles. The intersection of vertical stripe $i$ and horizontal stripe $j$ has color number $(i+j) \bmod 3$ for all $1 \leq i,j \leq n$. To prepare the frosting, Iskander wants to know the total surface in square centimeters to be colored for each of the three colors, and asks for your help.Input

The input consists of the following integers:
 - on the first line: the integer $n$,
 - on the second line: the values of $A_1,\dots ,A_ n$, $n$ integers separated with single spaces,
 - on the third line: the values of $B_1,\dots ,B_ n$, $n$ integers separated with single spaces.Limits

The input satisfies $3\leq n \leq 100000$ and $1 \leq A_1,\ldots ,A_ n,B_1,\ldots ,B_ n \leq 10000$.Output

The output should consist of three integers separated with single spaces, representing the total area for each color $0$, $1$, and $2$.

-----Examples-----
Sample Input:
3
1 1 1
1 1 1
Sample Output:
3 3 3
user: You just finished participating in a programming contest with your friend. Unfortunately, you were unable to All Kill the contest (i.e., solve all of the problems), but you are now wondering if there might be some strategy that would have solved all of the problems.

Solving a problem has two phases, a thinking phase and a coding phase. Your friend is responsible for all the thinking while you are responsible for all the coding.

For each problem, you’ve computed exactly how long it would take for you to code. However, before you can code a problem in contest, your friend needs to get the idea to solve it first. You aren’t sure how to estimate the time when your friend gets a solution idea, so you model it like this: For every problem, your friend gets the idea of how to solve this problem at a uniformly random minute during the contest. Each of these is an independent random variable. You can only code one problem at a time, so there may be several problems queued up at any moment of time. You always prioritize coding the lowest numbered problems first. You do this minute-by-minute, so you will switch to coding a lower-numbered problem if your friend gets the idea for it before you’re finished coding a higher-numbered problem, but you would prefer not to do this. Context switching is an expensive operation, even in the human brain!

The contest strategy can be modeled as follows for each minute:
 - For each problem that doesn’t yet have an idea, your friend will get the idea to solve it with probability $1/(\textrm{number of minutes remaining in contest})$. Your friend can get the idea to solve multiple problems in the same minute.
 - Among the problems that still need code time and your friend has gotten the solution idea, you will take the lowest numbered one and spend the next minute coding it (if no problem satisfies the condition, you do nothing at this step).

You would like to know the probability of these two events happening together:
 - Your team finishes coding all the problems by the end of the contest
 - For each problem, the time spent coding that problem is a contiguous interval

Let $p$ be this probability, $n$ be the number of problems in the contest and $t$ be the number of minutes in the contest. It can be shown that $p \cdot t^n$ is an integer. Output the value of $(p \cdot t^n) \pmod{998244353}$. Note that $998244353$ is a large prime.



-----Input-----
The first line of input contains two space-separated integers $n$ ($1 \leq n \leq 10^5$) and $t$ ($1 \leq t \leq 10^8$), where there were $n$ problems in the contest, and the contest lasted $t$ minutes.

Each of the next $n$ lines contains a single integer $x$ ($1 \leq x \leq 10^3$). These are the times to code each problem in minutes, in order. It is guaranteed that the sum of these times is less than or equal to $t$.

-----Output-----
Output a single integer, which is $(p \cdot t^n) \pmod{998244353}$, where $p$ is the probability of the two events mentioned above happening together.

-----Examples-----
Sample Input:
3 5
1
2
1
Sample Output:
60
user: You are given a collection of $N$ non-empty strings, denoted by $S_1, S_2, \ldots , S_ n$. Then you are given $N$-$1$ operations which you execute in the order they are given. The $i^{th}$ operation is has the following format: ‘$a$$b$’ ($1$-based indexing, without the quotes), which means that you have to make the following changes:
 - $S_ a = S_ a + S_ b$, i.e. concatenate $a^{th}$ string and $b^{th}$ string and store the result in $a^{th}$ string,
 - $S_ b$ = "", i.e. make the $b^{th}$ string empty, after doing the previous step.

You are ensured that after the $i^{th}$ operation, there will be no future operation that will be accessing $S_ b$. Given these operations to join strings, print the last string that will remain at the end of this process.

-----Input-----
The first line contains an integer $N$ ($1 \le N \le 10^5$) denoting the number of strings given. Each of the next $N$ lines contains a string denoting the $S_ i$. All the characters in the string $S_ i$ are lowercase alphabets from ‘a’ to ‘z’. The total number of characters over all the strings is at most $10^6$, i.e $\sum _{i = 1}^{N}|S_ i| \leq 10^6$, where $|S_ i|$ denotes the length of the $i^{th}$ string. After these $N$ strings, each of the next $N$-$1$ lines contain two integers $a$ and $b$, such that $a \neq b$ and $1 \le a, b \le N$ denoting the $i^{th}$ operation.

-----Output-----
Print the last string which remains at the end of the $N$-$1$ operations.

-----Warning-----
The I/O files are large. Please use fast I/O methods.

-----Examples-----
Sample Input:
4
cute
cat
kattis
is
3 2
4 1
3 4
Sample Output:
kattiscatiscute
user: Nathaniel collects rare gems and minerals, and has decided he has to protect his valuable collection from thieves. He is installing an electric eye system, in which he shines a laser into the room with his collection, and the laser will hit a sensor as long as no one steps in a spot that the laser crosses. Unfortunately, he spent most of his money on his collection, so he has only one laser and one sensor, and some mirrors. He has installed the mirrors at various locations in the room, at $45$ degree angles to the walls. At the point you have been called in to help, he has only one more mirror left to install, and he wants to know if there is any way to place it such that the laser will reach the sensor (as long as no one is in its way).

We will assume that the room is always square, the laser is always in the northern wall pointing south, and the sensor is mounted on the eastern wall, such that the laser must hit it from the west. We will also assume that the interior of the room can be represented by a grid, and that all mirrors (including the one to be added) are placed at the center of grid cells, at $45$ degrees with respect to the grid axis, and that the laser and sensor are at the center of their respective column/row of the grid.

-----Input-----
The first line of the input contains three integers: $n$, the size of the room, $c$, the column number of the laser ($1\leq c\leq n$), and $r$, the row number of the sensor ($1\leq r\leq n$). This is followed by $n$ lines, each containing $n$ characters separated by spaces. The character . represents an empty space, the character \ represents a mirror oriented NW/SE and the character / represents a mirror oriented NE/SW. You may assume $1\leq n\leq 2000$ and $1\leq r,c\leq n$.

-----Output-----
The output should consist of the string YES if the laser can be made to hit the sensor with the addition of at most one mirror, or the string NO otherwise.

-----Examples-----
Sample Input:
5 2 3
. . . . .
. . . . .
. . \ . .
. \ . . .
. . . . .
Sample Output:
NO
user: Gunnar and his friends like games which involve rolling dice. Gunnar has a huge collection of 6-sided, 12-sided and 20-sided dice. All the games with dice started to bore him, so he came up with a new game. He rolls an $s$-sided die $n$ times and wins if at least $k$ different numbers appear in the $n$ throws. An $s$-sided die contains $s$ distinct numbers $1, \dots , s$ on its sides.

Since this is a game only for one person, Gunnar and his friends decided to make it more fun by letting other people bet on a particular game. Before you bet on a particular game, you would like to know how probable it is to throw at least $k$ different numbers in $n$ throws with an $s$-sided die. We assume that all numbers have the same probability of being thrown in each throw.

-----Input-----
The input consists of a single line with three integers $n$, $s$, and $k$ ($1\le n\le 10000, 1 \le k \le s \le 500$). $n$ is the number of throws, $k$ the number of different numbers that are needed to win and $s$ is the number of sides the die has.

-----Output-----
Output one line with the probability that a player throws at least $k$ different numbers within $n$ throws with an $s$-sided die. Your answer should be within absolute or relative error at most $10^{-7}$.

-----Examples-----
Sample Input:
3 3 2
Sample Output:
0.888888889