meetforever/GRPO_CL huggingface dataset:
user: There is a heavy rainfall over the Antique Cultural Museum, and the roof has sprung a leak! All the beautiful paintings risk destruction, if it weren’t for one brave night watchman standing in the water’s way – Klunkas Plaskocek.

To protect all of the paintings, Klunkas has brought $N$ buckets and placed them in a long row to catch the water falling from the ceiling. Each bucket can hold exactly $K$ drops of water. If a drop falls into a bucket which is full, the water drop will overflow to the bucket standing to the left. If that bucket is full as well, the drop will keep flowing left until there is a non-full bucket.

Klunkas needs your help to prevent the leftmost bucket overflowing. After every drop of water falls, Klunkas wants to know which bucket it ended up in, so that he can keep track of the water levels of the buckets.

Due to the very regular nature of rain, Klunkas figured that the $R$ rain drops fall in a very nice pattern. Given constants $X_0, A, B$, we define the sequence $X_ i$ as\[ X_ i = (A \cdot X_{i-1} + B) \mod N \]

The bucket which the $i$-th rain drop falls into, is then $X_ i$ for $i = 1, \dots , R$. The leftmost bucket has the number $0$, and the rightmost bucket has the number $N-1$.

-----Input-----
Input consists of the space-separated integers $1 \le N \le 10^6$, $1 \le R \le 10^7$, $1 \le K \le 10$, $0 \le X_0, A, B \le 2^{31} - 1$.

-----Output-----
If the leftmost bucket overflows, output “OVERFLOW”.

Otherwise, the output should be calculated as follows:

Let $O_ i$ be the bucket which the $i$:th rain drop ended up in, and let $a = 0$.

Then, for every $i$ from $1$ to $R$, set $a := (53 \cdot a + O_ i) \mod 199933$.

The output should then be a line with the integer $a$.

-----Examples-----
Sample Input 1:
10 6 1 2 1 8
Sample Output 1:
OVERFLOW

Sample Input 2:
10 6 1 4 1 8
Sample Output 2:
79732
user: Most of us like playing video games. Benni prefers to play the video game Digbuild. Digbuild is primarily about surviving for as long as possible. In the game almost everything is possible. You can climb mountains, build castles, and fish, just to name a few options. The gameworld consists of large cubes, all the same size, whose corners are always in integral coordinates in three dimensional space. The player can both break these cubes (or blocks) and place blocks next to ones already there. There are also other items in the gameworld, auxiliary to these blocks. A few examples would be beds to sleep on, frames for photographs, and torches to light the world.

Benni isn’t a fan of building. He’d much rather dig tunnels in the ground. Benni always digs his tunnels horizontally and parallel to the $x$-axis of the gameworld. They are also always $3$ blocks high and $3$ blocks wide. Benni has just finished digging an $n$ block long tunnel and decided to go get a glass of water. When he sits down again to play some more he notices the tunnels are rather poorly lit. He realizes he has to place some torches on the floor of his tunnel to light the up. Benni is rather insistent on his tunnel not becoming ugly so he has to places the torches strategically. Benni considers his tunnel to be ugly if two blocks sharing a face both hold a torch.

In Digbuild you can only place one torch per block. Benni is so against his tunnel being ugly he’d rather have them unlit completely (i.e. not placing a torch is not considered ugly).

In how many different ways can Benni place the torches such that his tunnel doesn’t become ugly? Since this number may be rather large you are asked to find the answer $\mod 10^9 + 7$.

-----Input-----
The first and only line in the input contains the integer $1 \leq n \leq 10^{18}$.

-----Output-----
The only line in the output should contain the number of non-ugly torch arrangements in an $n$ block long tunnel, $\mod 10^9 + 7$.

-----Examples-----
Sample Input 1:
1
Sample Output 1:
5

Sample Input 2:
4
Sample Output 2:
227
user: Brandon Greg Jr. is moving to the United States to double his salary. He has $n$ items that he needs to pack into $k$ big boxes. The $n$ items are currently arranged in a row, and Brandon doesn’t want to bother reordering them, so he will partition the $n$ items into $k$ groups of consecutive items, and put each of the $k$ groups into their own box. For convenience when moving, Brandon wants to minimize the weight of the heaviest box. The weights of the boxes themselves are negligible.

-----Input-----
The first line contains two space-separated integers $n$ and $k$ ($1\le k\le n\le 10^5$), denoting the number of items and the number of boxes respectively.

The second line of input contains $n$ space-separated integers $w_ i$ ($1\le w_ i\le 10^4$), representing the weight of each item in order.

-----Output-----
The only line of output should contain a single integer, denoting the minimum possible weight of the heaviest box.

-----Examples-----
Sample Input 1:
7 2
3 1 1 3 9 5 2
Sample Output 1:
16

Sample Input 2:
7 4
1 2 8 3 5 2 7
Sample Output 2:
9
user: Mirko has received a homework assignment to compute the greatest common divisor of the two positive integers $A$ and $B$. Since the numbers are quite large, the teacher provided him with $N$ smaller integers whose product is $A$, and $M$ integers with product $B$.

Mirko would like to verify his result, so he has asked you to write a program to solve his problem. If the result is more than $9$ digits long, output only the last $9$ digits.

-----Input-----
The first line of input contains the positive integer $N$ ($1 \le N \le 1000$).

The second line of input contains $N$ space-separated positive integers less than $1000000000$, whose product is the number $A$.

The third line of input contains the positive integer $M$ ($1 \le M \le 1000$).

The fourth line of input contains $M$ space-separated positive integers less than $1000000000$, whose product is the number $B$.

-----Output-----
The first and only line of output must contain the greatest common divisor of numbers $A$ and $B$. If the result is more than $9$ digits long, output only the last (least significant) $9$ digits.

-----Examples-----
Sample Input 1:
3
2 3 5
2
4 5
Sample Output 1:
10

Sample Input 2:
4
6 2 3 4
1
1
Sample Output 2:
1
user: Angela is a new PhD student and she is nervous about the upcoming paper submission deadline of this year’s research conference. She has been working on multiple projects throughout the past year. Luckily most of the projects concluded successfully, and she came up with $n$ candidate papers. However not all of the papers were born equal—some have better results than others. Her advisor believes she should only submit the papers with “good enough” results so they have a high chance of getting accepted. 

Angela’s research group has a unique way of evaluating the success of paper submissions. They use the research productivity index, defined as $a^{a/s}$, where $s$ is the total number of papers submitted, and $a$ is the number of papers that are accepted by the conference. When $a = 0$, the index is defined to be zero. For example:
 - if one paper is submitted and it gets accepted, the index is $1^{1/1} = 1$;
 - if $4$ papers are submitted and all get accepted, the index is $4^{4/4} = 4$;
 - if $10$ papers are submitted and $3$ get accepted, the index is $3^{3/10} \approx 1.390389$;
 - if $5$ papers are submitted and $4$ get accepted, the index is $4^{4/5} \approx 3.031433$;
 - if $3$ papers are submitted and all get rejected ($a=0$), the index is $0$.

Intuitively, to get a high research productivity index one wants to get as many papers accepted as possible while keeping the acceptance rate high.

For each of her $n$ papers, Angela knows exactly how likely it is that the conference would accept the paper. If she chooses wisely which papers to submit, what is the maximum expected value of her research productivity index?

-----Input-----
The first line of the input has a single integer $n$ ($1 \leq n \leq 100$), the number of Angela’s candidate papers. The next line has $n$ space-separated integers giving the probability of each paper getting accepted. Each probability value is given as an integer percentage between $1$ and $100$, inclusive.

-----Output-----
Output the maximum expected value of Angela’s research productivity index. Your answer is considered correct if it has an absolute or relative error of no more than $10^{-6}$.

-----Examples-----
Sample Input 1:
5
30 50 70 60 90
Sample Output 1:
2.220889579

Sample Input 2:
6
30 90 30 90 30 90
Sample Output 2:
2.599738456
user: A new town is being built far out in the country, and currently there are $N$ houses. People have already started moving in. However, some of the houses aren’t connected to the internet yet, and naturally residents are outraged.

The houses are numbered $1$ to $N$. House number $1$ has already been connected to the internet via a long network cable to a neighboring town. The plan is to provide internet to other houses by connecting pairs of houses with separate network cables. A house is connected to the internet if it has a network cable to another house that’s already connected to the internet.

Given a list of which pairs of houses are already connected by a network cable, determine which houses are not yet connected to the internet.

-----Input-----
The first line of input contains two integers $1 \leq N,M \leq 200000$, where $N$ is the number of houses and $M$ is the number of network cables already deployed. Then follow $M$ lines, each containing a pair of distinct house numbers $1 \leq a, b \leq N$ meaning that house number $a$ and house number $b$ are already connected by a network cable. Each house pair is listed at most once in the input.

-----Output-----
If all the houses are already connected to the internet, output one line containing the string Connected. Otherwise, output a list of house numbers in increasing order, one per line, representing the houses that are not yet connected to the internet.

-----Examples-----
Sample Input 1:
6 4
1 2
2 3
3 4
5 6
Sample Output 1:
5
6

Sample Input 2:
2 1
2 1
Sample Output 2:
Connected
user: A molecule consists of atoms that are held together by chemical bonds. Each bond links two atoms together. Each atom may be linked to multiple other atoms, each with a separate chemical bond. All atoms in a molecule are connected to each other via chemical bonds, directly or indirectly.

The chemical properties of a molecule is determined by not only how pairs of atoms are connected by chemical bonds, but also the physical locations of the atoms within the molecule. Chemical bonds can pull atoms toward each other, so it is sometimes difficult to determine the location of the atoms given the complex interactions of all the chemical bonds in a molecule.

You are given the description of a molecule. Each chemical bond connects two distinct atoms, and there is at most one bond between each pair of atoms. The coordinates of some of the atoms are known and fixed, and the remaining atoms naturally move to the locations such that each atom is at the average of the locations of the connected neighboring atoms via chemical bonds. For simplicity, the atoms in the molecule are on the Cartesian $xy$-plane.

-----Input-----
The first line of input consists of two integers $n$ ($2 \leq n \leq 100$), the number of atoms, and $m$ ($n-1 \leq m \leq \frac{n(n-1)}{2}$), the number of chemical bonds.

The next $n$ lines describe the location of the atoms. The $i^\textrm {th}$ of which contains two integers $x, y$ ($0 \leq x,y \leq 1000$ or $x = y = -1$), which are the $x$ and $y$ coordinates of the $i^\textrm {th}$ atom. If both coordinates are $-1$, however, the location of this atom is not known.

The next $m$ lines describe the chemical bonds. The $i^\textrm {th}$ of which contains two integers $a$ and $b$ ($1 \leq a < b \leq n$) indicating that there is a chemical bond between atom $a$ and atom $b$.

It is guaranteed that at least one atom has its location fixed.

-----Output-----
Display $n$ lines that describe the final location of each atom. Specifically, on the $i^\textrm {th}$ such line, display two numbers $x$ and $y$, the final coordinates of the $i^\textrm {th}$ atom. If there are multiple solutions, any of them is accepted. A solution is accepted if the coordinates of each unknown atom and the average coordinates of all its neighboring atoms via chemical bonds differ by at most $10^{-3}$. Note that it is acceptable for multiple atoms to share the same coordinates.

-----Examples-----
Sample Input 1:
3 2
0 0
-1 -1
2 0
1 2
2 3
Sample Output 1:
0 0
1 0
2 0

Sample Input 2:
5 4
0 0
-1 -1
-1 -1
-1 -1
4 0
1 2
2 3
3 4
4 5
Sample Output 2:
0 0
1 0
2 0
3 0
4 0
user: Your friend has gotten lost in the woods. He has called and asked for you to come get him, but you are very busy and would rather just stay home. You quickly look up a map of the woods. It appears that the woods consist of a small number of clearings, with paths connecting them. You hope that the woods are sufficiently small and simple that your friend can get out easily, even if he is just randomly running around.

From your friend’s description, you can figure out at which clearing he is. Assuming that every time he reaches a clearing, he runs in a uniformly random direction (including back the way he came), and that it takes him exactly one minute to get from clearing to clearing, can you predict how long it will take him to get out on average?

-----Input-----
The first line contains two integers $N$ and $M$, where $N$ is the number of clearings in the woods ($2 \leq N \leq 20$), and $M$ is the total number of paths between clearings. The clearings are numbered $0$ through $N-1$, such that clearing $0$ is the one where your friend is right now and clearing $N-1$ is the exit of the woods.

The next $M$ lines each contain two integers $K$ and $L$, indicating a path between clearing $K$ and clearing $L$ ($0 \leq K, L < N$, $K \neq L$).

You may assume that it is possible for your friend to reach the exit by following paths, that paths do not cross, and that there is at most one path between any two clearings.

-----Output-----
Output a single line containing a single number: the expected value of the number of minutes it will take your friend to get out of the woods.

Your answer may have an absolute error of at most $10^{-5}$.

-----Examples-----
Sample Input 1:
3 3
0 1
1 2
0 2
Sample Output 1:
2.000000

Sample Input 2:
5 6
0 1
0 2
1 2
2 4
0 3
3 4
Sample Output 2:
6.727273
user: Biologists have discovered a strange DNA molecule, best described as a sequence of $N$ characters from the set $\{ A, B\} $. An unlikely sequence of mutations has resulted in a DNA strand consisting only of $A$’s. Biologists found that very odd, so they began studying the mutations in greater detail.

They discovered two types of mutations. One type results in changing any single character of the sequence ($A \rightarrow B$ or $B \rightarrow A$). The second type changes a whole prefix of the sequence, specifically replacing all characters in positions from $1$ to $K$ (for some $K$ between $1$ and $N$, inclusive) with the other character ($A$ with $B$, $B$ with $A$).

Compute the least possible number of mutations that could convert the starting molecule to its end state (containing only $A$ characters). Mutations can occur in any order.

-----Input-----
The first line of input contains the positive integer $N$ ($1 \le N \le 1000000$), the length of the molecule. The second line of input contains a string with $N$ characters, with each character being either $A$ or $B$. This string represents the starting state of the molecule.

-----Output-----
The first and only line of output must contain the required minimum number of mutations.

-----Examples-----
Sample Input 1:
4
ABBA
Sample Output 1:
2

Sample Input 2:
5
BBABB
Sample Output 2:
2
user: $N$ points are placed in the coordinate plane.

Write a program that calculates how many ways we can choose three points so that they form a right triangle with legs parallel to the coordinate axes.

A right triangle has one 90-degree internal angle. The legs of a right triangle are its two shorter sides.

-----Input-----
The first line of input contains the integer $N$ ($3 \le N \le 100000$), the number of points.

Each of the following $N$ lines contains two integers $X$ and $Y$ ($1 \le X, Y \le 100000$), the coordinates of one point.

No pair of points will share the same pair of coordinates.

-----Output-----
Output the number of triangles.

-----Examples-----
Sample Input 1:
3
4 2
2 1
1 3
Sample Output 1:
0

Sample Input 2:
5
1 2
2 1
2 2
2 3
3 2
Sample Output 2:
4
user: Mafia is a social game played frequently by high school competitors in informatics on summer and winter camps and national competitions, usually very late at night, drinking various fruit sodas. This game is not about winning, it’s about los taking part, like in competitions.

To solve this task, you don’t need to know the rules of mafia: all you need to know is that some of the players are “mobsters” and the rest are “civilians”. The mobsters know who is who, but the civilians don’t. The civilians are trying to figure out who the mobsters are during the game.

In the current round of the game, out of $N$ surviving players so far, each one has accused exactly one other player saying that he is the mobster. The civilians were only guessing and the mobsters have accused civilians, pretending to know nothing.

Not knowing who the mobsters are, but knowing who accused whom, determine the maximum possible number of mobsters among these players!

-----Input-----
The first line of input contains the integer $N$ ($2 \leq N \leq 500000$), the number of players. The players are labeled with integers from $1$ to $N$.

The $K$-th line of input, out of the following $N$ lines, contains the label of the player accused by the player $K$. (No player can accuse themselves).

-----Output-----
The first and only line of output must contain the maximum possible number of mobsters.

-----Examples-----
Sample Input 1:
3
2
1
1
Sample Output 1:
2

Sample Input 2:
3
2
3
1
Sample Output 2:
1
user: Particle colliders are difficult to build and experiments are costly to run. Before running any real experiments it is better to do a simulation to test out the ideas first. You are required to write a very simple simulator for this problem.

There are only three particles in this system, and all particles are confined to an infinite plane so that they can be modelled as circles. Their locations are specified only by the $x_ i$ and $y_ i$ coordinates of their centers ($1 \leq i \leq 3$). All three particles have the same radius $r$, and are initially stationary.

We are given a vector $(x_ v, y_ v)$ specifying the direction particle $1$ will move when the experiment starts. When particle $i$ hits particle $j$, particle $j$ will start moving in the direction perpendicular to the tangent at the point of the contact, away from particle $i$. Particle $i$ will cease to exist and be converted to radiation. A moving particle that does not hit another will continue moving indefinitely.

There are a number of possible scenarios your simulator should identify:
 - particle $1$ hits particle $2$, which in turns hits particle $3$;
 - particle $1$ hits particle $3$, which in turns hits particle $2$;
 - particle $1$ hits particle $2$, which moves indefinitely;
 - particle $1$ hits particle $3$, which moves indefinitely;
 - particle $1$ moves indefinitely.

-----Input-----
The input contains four lines. The first three lines each contains two integers $x_ i$ and $y_ i$ ($|x_ i|,|y_ i| \leq 1000$), describing particles $1$, $2$, and $3$ in this order. The fourth line contains three integers $x_ v$, $y_ v$, and $r$ ($|x_ v|, |y_ v| \leq 1000$, $0 < r \leq 50$).

You may assume that no two particles touch or overlap initially, and that the distance between the centers of particles $2$ and $3$ is greater than $4r$.

-----Output-----
Output a single integer giving the number ($1$–$5$) identifying the scenarios described above.

Although you should take care of your calculations, it is guaranteed that the outcome would not change if the initial vector $(x_ v,y_ v)$ is rotated by one degree either way.

-----Examples-----
Sample Input 1:
0 0
50 45
91 50
42 50 10
Sample Output 1:
1

Sample Input 2:
0 0
50 50
141 50
41 50 10
Sample Output 2:
3
user: You are a stone-chiseler in Ancient Greece, charged with copying the text from famous monuments onto new stones. The existing inscriptions use boustrophedon writing, in which the lines of text alternate direction, and if a line ends in the middle of a word, the word simply continues on the next line. For example, an inscription may look like this:

IN.HONOR.OF.THE.INTERNAT
TEPMOC.GNIMMARGORP.LANOI
ITION

(In the original boustrophedon style, no spaces or punctuation were used between words, but here we have introduced periods between words.)

Today, however, you have been asked to copy these over into the new-fangled style where each line runs left to right, and words will not be split across lines — if there is not room for a whole word on a particular line, it will be started on the next line. So, the inscription above might instead look like this:

IN.HONOR.OF.THE
INTERNATIONAL
PROGRAMMING.COMPETITION

Of course, before you get started, you will need to know what size of stone will be required for your inscription. The width of the stone will be given, and the length for you to determine.

-----Input-----
The first line of the input will consist of three numbers, $n$, the number of lines in the original inscription, $d$, the width of the original inscription, and $w$, the width of your new stone in characters. Then, $n$ lines follow containing the original inscription written in boustrophedon style. You may assume that the first $n-1$ such lines consist only of capital letters and periods, with the $n$-th line also possibly containing spaces. You may assume that no two periods appear consecutively, that $1\leq n \leq 100000$, $1\leq d\leq 100$, $1\leq w\leq 100$, and that no word is more than $w$ letters long.

-----Output-----
The output should consist of a single integer, representing the number of lines required to write the inscription when writing left to right and not splitting words across lines.

-----Examples-----
Sample Input 1:
3 24 24
IN.HONOR.OF.THE.INTERNAT
TEPMOC.GNIMMARGORP.LANOI
ITION
Sample Output 1:
3

Sample Input 2:
3 24 20
IN.HONOR.OF.THE.INTERNAT
TEPMOC.GNIMMARGORP.LANOI
ITION
Sample Output 2:
4