meetforever/GRPO_CL huggingface dataset:
user: One fateful spring evening Bergur was playing his favorite video game Striker-Count. Bergur is so good at Striker-Count he usually defeats his opponents without breaking a sweat. Therefore he’s been racking his brain for new ways to win with style. He decides to jump in the air, rotate a full circle and shoot one shot with his trusty PWA. But he’s been wondering, what is the maximum amount of opponents he can hit with one shot in this fashion.

We consider Bergur to positioned at the origin of a plane. His opponents are then provided as circles with a center somewhere in the plane and a radius. A single shot is then a ray from the origin and hits the opponent if and only if the ray intersects strictly more than one point of the opponent (i.e. grazing the opponent doesn’t count as a hit). It is guaranteed that no opponents include Bergur (the origin). Enemies might overlap however since they might be standing on top of one another. By sheer skill Bergur will still manage to hit both of them with a single shot if the ray intersects both circles.

-----Input-----
The first line of the input contains an integer $1 \leq n \leq 10^5$, which denotes the number Bergur’s opponents present on the plane. Then $n$ lines follow, the $i$th of which contains three real numbers $-10^9 \leq x_ i, y_ i \leq 10^9$, $0 < r \leq 10^9$ which denote an opponent centered at point $(x_ i, y_ i)$ in the plane with radius $r_ i$. All real numbers will have at most six digits after the decimal point. It is guaranteed that there is a neighbourhood around an optimal angle of radius $10^{-9}$ such that all angles in the neighbourhood give the same number of hit enemies.

-----Output-----
The only line of the output should contain the maximum number of opponents Bergur can hit with one shot.

-----Examples-----
Sample Input:
5
5 0 1
10 0 1
0 5 1
0 -5 1
-5 0 1
Sample Output:
2
user: Lukáš really likes orienteering, a sport that requires locating control points in rough terrain. To entertain the NWERC participants Lukáš wants to organize an orienteering race. However, it would be too harsh for the participants to be outdoors in this cold Swedish November weather, so he decided to jump on the new trend of indoor races, and set the race inside the B building of Linköping University.

Lukáš has already decided on the locations of the control points. He has also decided on the exact length of the race, so the only thing remaining is to decide in which order the control points should be visited such that the length of the total race is as he wishes. Because this is not always possible, he asks you to write a program to help him.

Note from the organizer: the NWERC indoorienteering race for this year has been cancelled since we neglected to apply for an orienteering permit in time from the university administration. (We still need you to solve the problem so that we can organize it for next year.)

-----Input-----
The input consists of:
 - one line with two integers $n$ ($2\leq n \leq 14$) and $L$ ($1 \leq L \leq 10^{15}$), the number of control points and the desired length of the race, respectively;
 - $n$ lines with $n$ integers each. The $j$th integer on the $i$th line, $d_{ij}$, denotes the distance between control point $i$ and $j$ ($1\leq d_{ij} \leq L$ for $i \not= j$, and $d_{ii} = 0$). For all $1 \leq i,j,k \leq N$ it is the case that $d_{ij} = d_{ji}$ and $d_{ij} \leq d_{ik} + d_{kj}$.

-----Output-----
Output one line with “possible” if it is possible to visit all control points once in some order and directly return to the first one such that the total distance is exactly $L$, and “impossible” otherwise.

-----Examples-----
Sample Input:
4 10
0 3 2 1
3 0 1 3
2 1 0 2
1 3 2 0
Sample Output:
possible
user: Cesar and Raul like betting and good food, in no particular order. They want to try out a new fancy restaurant and they decided to make a bet – they are going to play a game and the loser pays for dinner.

They have a box with $N$ balls. Each ball contains a distinct number between $1$ and $N$. Then, the game proceeds with these steps:
 - Initially, each person picks $C$ distinct numbers between $1$ and $N$ and writes them down on a separate card.
 - In each round, $D$ balls are drawn from the box uniformly at random. Cesar and Raul mark the ball numbers that appear on their respective card. The $D$ balls are then returned to the box.
 - The game stops when a player has marked all the numbers he chose. That player is the winner. If both players finish at the same time, it is a draw and they will split the dinner.

They are quite eager to try out this new restaurant and they’re now wondering: how many rounds will the game last until at least one of the players has marked all his numbers?

-----Task-----
Given the number $N$ of balls, the number $D$ of balls they draw from the box in each round, the amount $C$ of numbers in their cards and the numbers they wrote down, find the expected number of rounds the game will last.

-----Input-----
The first line of the input consists of three space separated integers: $N$, $D$, and $C$. $N$ is the number of balls, $D$ is the number of balls drawn in each round, and $C$ is the cards’ size. Each of the following two lines contains $C$ space separated integers: the numbers Cesar and Raul wrote down, respectively.

-----Constraints-----

$1$

$\leq $

$N$

$\leq $

$50$

Number of balls in the box

$1$

$\leq $

$D$

$\leq $

$\min (10, N)$

Number of balls drawn in each round

$1$

$\leq $

$C$

$\leq $

$\min (10, N)$

Cards’ sizeOutput

The output is the expected number of rounds of the game.

The result will be considered correct as long as the absolute error does not exceed $10^{-3}$.

-----Explanation for Sample Input 1-----
There are $2$ balls. Cesar picked number $1$ and Raul picked number $2$. In the first round, either number $1$ or $2$ will be drawn and so one of them wins right away.

-----Examples-----
Sample Input:
2 1 1
1
2
Sample Output:
1.00000000
user: Young Luka is about to enter a house with the evil witch Marica inside. As soon as he enters the house, she asks him questions about her array of $N$ numbers. Luka fearfully asks for a clarification of the questions. Marica explains to him that each query consists of two integers $L$ and $R$ which represent the positions of a contiguous sub-array in her array.

It is Luka’s task to answer for each query what the longest contiguous sub-array of that contiguous sub-array (it can be the entire sub-array) having the property of being magical. An array is called magical if all the values are between the values of the first and last number in that array. For example, $[1\ 3\ 1\ 2\ 4]$ is magical, the same as $[4\ 1\ 1\ 2\ 1]$, whereas $[3\ 3\ 4\ 1]$ is not magical.

-----Input-----
The first line of input contains the integer $N$ ($1 \leq N \leq 500000$), the number of numbers in the array. The second line contains $N$ integers $a_ i$ ($1 \leq a_ i \leq 10^9$). The third line contains the integer $Q$ ($1 \leq Q \leq 500000$), the number of queries. Each of the following $Q$ lines contains two integers, $L$ and $R$ ($1 \leq L \leq R \leq N$), representing the sub-array from the query.

-----Output-----
The $i$-th line of output must contain a single integer—the answer to the $i$-th query.

-----Examples-----
Sample Input:
5
5 4 3 3 2
3
1 2
1 1
2 4
Sample Output:
2
1
3
user: MTR (subway) stations are common meetup locations in Hong Kong. Alice and Bob are going to meet up at one of the MTR stations. They each show up at the meeting spot in one of the stations right at noon, without remembering which station they plan to meet at. Having poor cellphone connection at the stations, they fail to contact the other person. Fortunately, they have agreed to look for each other with the following strategy.

Each station has a unique designated meeting spot. Every minute starting from noon, Alice and Bob look for the other person at the meeting spot of their current stations. If they happen to be at the same station, they can finally meet up. Otherwise, they each takes a train to reach a neighbouring station, uniformly at random from among the neighbouring stations. Upon arrival, they look for the other person at the meeting spot of their current stations. If they still cannot find each other, each of them again takes a train to reach a neighbouring station, uniformly at random. This process goes on until they are at (the meeting spot of) the same station at the same time.

Trains move very fast, so Alice and Bob will not see each other while on trains running in opposite directions. Suppose it takes exactly one minute to get from the meeting spot of one station to that of a neighbouring station. What is the expected time they will meet?

-----Input-----
The first line of input contains two integers $n$ and $m$. $n$ is the number of MTR stations ($1 \leq n \leq 20$, $0 \leq m \leq n(n-1)/2$) and $m$ is the number of pairs of stations that are neighbours.

Each of the following $m$ lines contains two distinct integers, $u$ and $v$ ($0 \leq u, v < n$, $u \neq v$), indicating that stations $u$ and $v$ are neighbours. Every unordered pair of neighbours $(u,v)$ will appear at most once in the input.

The last line consists of two integers $s$ and $t$ ($0 \leq s, t < n$), denoting the initial stations of Alice and Bob, respectively.

-----Output-----
If Alice and Bob will never meet, output “never meet” without the quotes. Otherwise, output a real number indicating the expected time (in minutes past noon) they will meet at the meeting spot of a certain station. Any solution with a relative or absolute error of $10^{-6}$ will be accepted.

-----Examples-----
Sample Input:
3 2
0 1
1 2
0 2
Sample Output:
1
user: “The power to arrest­–to deprive a citizen of liberty­–must be used fairly, responsibly, and without bias.” ­–Loretta Lynch.

Due to desperation caused by the global financial crisis, you, left with no other options, decided to rob a bank to survive. Unfortunately, while you were able to escape with a small fortune, the city’s police is now out in full force trying to hunt you down.

The city can be represented as a grid with $R$ rows and $C$ columns. Each block in the city is represented as a cell in this grid. The city has very strict border controls and has shut down all its borders for $24$ hours after the bank robbery to find and arrest you.

You have intercepted the officer’s calls and have discovered that the police force intends to, for the next $24$ hours, search all blocks within $K$ blocks of their current location (that is, all blocks that can be reached by traveling a total of at most $K$ blocks horizontally or vertically). Unfortunately, you don’t know their current location, so you can only assume that they are at some random block, each block with equal probability.

You hence also intend to hide at a random location in the city. You will choose a block uniformly at random and hide there, hoping that it is not a block the police will search. Since you have no idea where the police are, this may unfortunately also be a block the police is currently at (and will definitely search).

You can assume that, unfortunately, the police are very thorough in their search and they will definitely catch you if they search the block you are in. However, if the $24$ hours elapse without the block you are in being searched, you will surely be able to escape as the borders are reopened.

What is the probability that you will evade capture?

-----Input-----
The first and only line of input contains three integers, $R$, $C$ ($1 \leq R, C \leq 10^{11}$) and $K$ ($1 \leq K \leq 2\cdot 10^{11}$), the number of rows in the city, the number of columns in the city and the number of blocks the police will search, respectively.

-----Output-----
It can be shown that the probability of evading capture can be uniquely represented as an irreducible fraction $p/q$.

Output a single integer on a line by itself, containing the remainder after dividing $p\cdot q^{-1}$ by $10^{11}+3$, where $q^{-1}$ denotes the modular multiplicative inverse of $q$ with respect to the modulus $10^{11}+3$.

The input will be such that $q^{-1}$ always exists.

-----Examples-----
Sample Input:
5 4 2
Sample Output:
14500000001
user: On a faraway planet, strange plants with two stems can be found. Every plant on the planet can be described by three numbers: the $x$-coordinates $L$ and $R$ of the two stems, and the height $H$ at which the stems are connected. The following image depicts a plant with $L=2$, $R=5$ and $H=4$.

Every day a new plant grows on the planet. The plant that grows on day $1$ is of height $1$, and every subsequent plant is one higher than the previous one.

When a stem of a new plant intersects the horizontal segment of another plant, a small flower grows (if one was not there already). If segments merely touch in a point, a flower will not grow there.

The following images are a visualization of the first sample input.         

Write a program that, given the coordinates of all plants, calculates the number of new flower every day.

-----Input-----
The first line contains an integer $N$ ($1 \le N \le 100000$), the number of days.

Each of the following $N$ lines contains two integers $L$ and $R$ ($1 \le L < R \le 100000$). The $i$’th of these lines describes the coordinates of the plant that grows on day $i$.

-----Output-----
Output $N$ lines, the number of new flowers after each plant grows.

-----Examples-----
Sample Input:
4
1 4
3 7
1 6
2 6
Sample Output:
0
1
1
2
user: Note that this is a bit harder version of the problem foodreview.

Frida is a writer for Cosmopolitan who writes restaurant reviews. She enjoys it a lot, but it seems that, throughout the years, she has reviewed all the restaurants on Earth. It’s now time to move one level up; she is going to review the food served by the airlines, so that the readers can make better decisions on which flights to take.

Her boss gave her a list of flight connections that she needs to review for the upcoming issue of Cosmopolitan. She knows that they serve the same food in both directions of every flight, so she only needs to take it once. She realized that she will need to take some additional flights, because she can not make all reviews using only flights in the list from her boss. Therefore she did some quick research and made a list of additional flights which she might take. She will not review the food on these flights; they will only be used so that she can make all the reviews.

Frida’s goal is to make all the reviews while spending the least money on flight tickets. Her office is in Stockholm, so she starts and ends her journey there. Each flight is both ways between two cities and has a fixed price in both directions. You can assume that it is possible to make all the reviews using some of the additional flights.

For the purposes of this problem we ignore the price Frida has to pay for accommodation and we also ignore the departure and arrival times of flights by assuming that every flight is very often and reasonably short. We only focus on the total price of the flights.

-----Input-----
The first line contains $2$ space separated integers $N, R, (2\leq N\leq 15, 0\leq R\leq 105)$, where $N$ is the number of airports mentioned in the input and $R$ is the number of flights to review. The airports are numbered $1, \dots , N$ and Stockholm has number $1$.

The next $R$ lines describe the $R$ flights to review. Each line contains 3 space separated integers $a, b, c, (1\leq a, b\leq N, 1\leq c\leq 10000)$, where $a, b$ denote 2 distinct airports and $c$ is the cost of the flight in Swedish kronor in both directions. No pair of $2$ cities is listed twice.

The next line contains an integer $F$, $(0\leq F\leq 250)$, the number of additional flights available. The next $F$ lines contain descriptions of flights in the same format as above. There may be more flights between a pair of cities. You may assume that it is possible to make all the reviews using some of these additional flights.

-----Output-----
Output one line with one integer – the lowest total cost of flight tickets, such that Frida can make all the reviews and return back to Stockholm.

-----Examples-----
Sample Input:
5 3
1 2 1000
2 3 1000
4 5 500
2
1 4 300
3 5 300
Sample Output:
3100
user: Statisticians like to create a lot of statistics. One simple measure is the mean value: the sum of all values divided by the number of values. Another is the median: the middle among all values when they have been sorted. If there are an even number of values, the mean of the two middle values will form the median.

These kinds of measures can be used for example to describe the population in a country or even some parts of the population in the country. Anne Jensen, Maria Virtanen, Jan Hansen, Erik Johansson and Jón Þórsson want to find a statistical measurement of how many statisticians there are in the Nordic countries. To be more precise, they want to find out how many statisticians there are per unit area. As the population in the Nordic countries are well spread out they will try the new measurement MAD, Median of All Densities. First put a square grid on the map. Then draw a rectangle aligned with the grid and calculate the density of statisticians in that area, i.e. the mean number of statisticians per area unit. After that, repeat the procedure until all possible rectangles have been covered. Finally the MAD is the median of all statistician densities.

-----Input-----
The first line of the input contains of two space separated numbers $h$ and $w$ describing the height and width of the square grid, where $1 \leq h \leq 140$ and $1 \leq w \leq 120$. The next line contains two space separated numbers $a$ and $b$ which are the lower and upper bound of the allowed rectangle areas, i.e. $1 \leq a \leq rectangle area \leq b \leq w \times h$. Then there will follow $h$ lines with $w$ space separated numbers $s$ describing the number of statisticians in each square of the map, $0 \leq s \leq 10000$. There will always exist a rectangle with an area in $[a,b]$.

-----Output-----
The output contains of one line with the MAD. The number should be printed in number of statisticians per square and have absolute error at most $< 10^{-3}$.

-----Examples-----
Sample Input:
4 2
1 8
6 5
2 5
2 9
7 13
Sample Output:
5.250000000
user: The problem is simple. You are given a long sequence of integers $a_1, a_2, \ldots , a_ n$. Then you are given a query consisting of a start index $i$ and a subset of integers $B$. What is the longest consecutive subsequence of the given sequence that starts at position $i$ and contains only integers in $B$?

Simple, right?

-----Input-----
The first line of input contains a single integer $1 \leq n \leq 10^5$. The second line contains $n$ integers $a_1, \ldots , a_ n$. Each integer $a_ j$ lies between 0 and $2^{31}-1$.

The third line contains a single integer $q \geq 1$ indicating the number of queries to process. Then $q$ lines follow, each starting with two integers $1 \leq i \leq n$ and $1 \leq m \leq 10^5$, followed by $m$ distinct integers $b_1, \ldots , b_ m$. Each integer $b_ j$ lies between 0 and $2^{31}-1$.

Finally, you are guaranteed the sum of all values $m$ over all queries is at most $10^5$.

-----Output-----
For each query, output a single line with the length of the longest prefix of $a_ i, a_{i+1}, \ldots , a_ n$ that only contains integers from $B$.

-----Examples-----
Sample Input:
7
1 2 3 1 2 1 1
5
1 3 1 2 3
1 2 1 2
2 2 2 3
3 2 1 2
4 2 1 2
Sample Output:
7
2
2
0
4
user: Have you recently visited an arcade? Arcade games seem to have become more boring over the years, requiring less and less skill. In fact, most arcade games these days seem to depend entirely on luck. Consider the arcade game shown in the picture, which consists of different holes arranged in a triangular shape. A ball is dropped near the hole at the top. The ball either falls into the hole, in which case the game ends, or it bounces to one of its (up to) $4$ neighbors, denoted by the red arrows. Different holes have different payouts — some may even be negative! If the ball reaches another hole, the process repeats: the ball either falls into the hole, ending the game — or it bounces to one of its neighbors, possibly ad infinitum! 

Write a program that computes the expected payout when dropping a ball into the machine!

-----Input-----
The input consists of a single test case. The first line contains an integer $N$ ($1 \le N \le 32$) describing the number of rows of the arcade machine. The second line contains $H = N (N+1) / 2$ integers $v_ i$ ($-100 \le v_ i \le 100$) describing the payout (positive or negative) if the ball drops into hole $i$. Holes are numbered such that hole $1$ is in the first row, holes $2$ and $3$ are in the second row, etc. The $k^{\textrm{th}}$ row starts with hole number $k (k-1) / 2 + 1$ and contains exactly $k$ holes.

These two lines are followed by $H$ lines, each of which contains $5$ real numbers $p_0 \ p_1 \ p_2 \ p_3 \ p_4$, denoting the probability that the ball bounces to its top-left ($p_0$), top-right ($p_1$), bottom-left ($p_2$), or bottom-right ($p_3$) neighbors or that the ball enters the hole ($p_4$). Each probability is given with at most $3$ decimal digits after the period. It is guaranteed that $0.0 \le p_ i \le 1.0$ and $\sum p_ i = 1.0$. If a hole does not have certain neighbors because it is located near the boundary of the arcade machine, the probability of bouncing to these non-existent neighbors is always zero. For instance, for hole number $1$, the probabilities to jump to the top-left and top-right neighbors are both given as $0.0$.

You can assume that after the ball has bounced $b$ times, the probability that it has not fallen into a hole is at most $(1 - 10^{-3})^{\lfloor b/H \rfloor }$.

-----Output-----
Output a single number, the expected value from playing one game. Your answer is considered correct if its absolute or relative error is less than $10^{-4}$.

Hint: Using Monte Carlo-style simulation (throwing many balls in the machine and simulating which hole they fall into using randomly generated choices) does not yield the required accuracy!

-----Examples-----
Sample Input:
4
40 30 30 40 20 40 50 30 30 50
0.0 0.0 0.45 0.45 0.1
0.0 0.3 0.3 0.3 0.1
0.3 0.0 0.3 0.3 0.1
0.0 0.3 0.3 0.3 0.1
0.2 0.2 0.2 0.2 0.2
0.3 0.0 0.3 0.3 0.1
0.0 0.8 0.0 0.0 0.2
0.4 0.4 0.0 0.0 0.2
0.4 0.4 0.0 0.0 0.2
0.8 0.0 0.0 0.0 0.2
Sample Output:
32.6405451448