meetforever/GRPO_CL huggingface dataset:
user: After finding a suitable residence in the Swiss Alps, Robin thought he finally had what it takes to be happy. But every day he woke up feeling that something was missing. Maybe it was the lack of English food in these parts? Spotting a market opportunity and solving his problem at the same time, he teamed up with the famous English chef Jim to open a restaurant nearby. While he has no doubts about Jim’s talents as a chef, Robin wants to be sure that opening an English restaurant in the Swiss Alps is a good idea.

Fortunately, as a local, he knows the habits of restaurant customers. At the stroke of each hour, a group of people arrives of size that is uniformly random between $1$ and $g$ people (inclusive). The group finds the smallest completely unoccupied table that fits the group and occupies it. If they can not find such a table, the group leaves with great disappointment. Once seated, a group never leaves until the restaurant closes, as Jim has no difficulty keeping the guests entertained.

As an example, suppose the restaurant has $3$ tables of capacities $5$, $8$ and $9$. If groups of sizes $5, 10$ and $3$ arrive (in that order), in the end there will be $8$ people in the restaurant. The first group occupies the table of capacity $5$, the second group leaves and the last group occupies the table of capacity $8$.

Robin plans to keep his restaurant open for $t$ hours in total. In the restaurant business the most important metric is the expected number of people in the restaurant when it is closes. Can you help Robin calculate the expected occupancy after $t$ hours?

-----Input-----
The input consists of:
 - One line with three integers $n, g, t$ ($1 \le n \le 100, 1\leq g\leq 200, 1\leq t \leq 100$), the number of tables in the restaurant, the maximum group size, and the number of hours the restaurant is open.
 - One line with $n$ integers $c_1, \ldots , c_ n$ ($1 \le c_ i \le 200$ for each $i$) giving the capacities of the tables.

-----Output-----
Output the expected number of people in the restaurant when it closes. Your answer should have an absolute or relative error of at most $10^{-6}$.

-----Examples-----
Sample Input 1:
3 3 2
1 2 3
Sample Output 1:
3.666666667

Sample Input 2:
4 11 4
10 10 10 10
Sample Output 2:
20.000000000
user: Everybody loves big numbers (if you do not, you might want to stop reading at this point). There are many ways of constructing really big numbers known to humankind, for instance: 
 - Exponentiation: $42^{2016} = \underbrace{42 \cdot 42 \cdot \ldots \cdot 42}_{2016\text { times}}$.
 - Factorials: $2016! = 2016 \cdot 2015 \cdot \ldots \cdot 2 \cdot 1$.

In this problem we look at their lesser-known love-child the exponial, which is an operation defined for all positive integers $n$ as\[ \operatorname {exponial}(n) = n^{(n-1)^{(n-2)^{\cdots ^{2^{1}}}}}. \]

For example, $\operatorname {exponial}(1) = 1$ and $\operatorname {exponial}(5) = 5^{4^{3^{2^1}}} \approx 6.206 \cdot 10^{183230}$ which is already pretty big. Note that exponentiation is right-associative: $a^{b^c} = a^{(b^c)}$.

Since the exponials are really big, they can be a bit unwieldy to work with. Therefore we would like you to write a program which computes $\operatorname {exponial}(n) \bmod m$ (the remainder of $\operatorname {exponial}(n)$ when dividing by $m$).

-----Input-----
The input consists of two integers $n$ ($1 \le n \le 10^9$) and $m$ ($1 \le m \le 10^9$).

-----Output-----
Output a single integer, the value of $\operatorname {exponial}(n) \bmod m$.

-----Examples-----
Sample Input 1:
2 42
Sample Output 1:
2

Sample Input 2:
5 123456789
Sample Output 2:
16317634
user: In the hit video game “Johnny5 and the Exploding Oil Cans”, you control the robot “Johnny5”. The robot can be moved one cell at a time in one of the directions up, down, left, or right on an $N \times N$ grid. One move takes one second and costs one unit of energy. It is allowed to stand still, you don’t have to move every second. If you have no energy, you can’t move. You start the game with $E$ units of energy.

The objective of the game is to collect as many oil cans as you can. Every second there may appear one or more oil cans somewhere on the grid. If Johnny5 is at a location when a can appears there he collects that can, otherwise it explodes. You score $1$ point for each oil can Johnny5 collects. If an oil can explodes in one of the four adjacent cells to where Johnny5 is located he collects the spilled oil, and you gain one unit of energy for each of them. If he does not pick up the can, and is not in one of the adjacent cells to pick up the oil, the oil disappears immediately. Note that he only gets oil from adjacent cells, and not from any cans in the same cell that he is in.

You’ve had trouble beating this game for years, but your friend just called and told you there is a way to get a list of where and when the cans will appear. Write a program that uses this information to find the maximum number of points you can get.

-----Input-----
The first line of the input consists of $5$ space-separated integers $N$, $E$, $S_ X$, $S_ Y$, $C$. These numbers give the size of the grid, the starting energy, the $x$ and $y$ coordinates where Johnny5 starts, and the number of cans.

The next $C$ lines each consist of $3$ space-separated integers $X$, $Y$, $CT$. These numbers represent the $x$ and $y$ coordinates of a can, and the time it appears there, in seconds after the start of the game.

-----Output-----
Output the maximum number of points you can score.

-----Limits-----
 - $1 \leq N \leq 500$.
 - $0 \leq E \leq 100$.
 - $0 \leq C \leq 100$.
 - $0 \leq X_ S, Y_ S, X, Y < N$.
 - $1 \leq CT \leq 100$.

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

Sample Input 2:
3 1 1 1 8
0 1 1
1 0 1
2 1 1
1 2 1
1 2 2
2 2 3
0 2 5
1 2 6
Sample Output 2:
4
user: You are playing a racing game. Your character starts at the X-axis line ($y=0$) and proceeds up the racetrack, which has a boundary at the lines $x=0$ and $x=w$. The finish is at $y=h$, and the game ends when you reach that line. You proceed at a fixed vertical velocity $v$, but you can control your horizontal velocity to be any value between $-v/r$ and $v/r$, where $r$ is a fixed ratio. You may change your horizontal velocity at any time, but your vertical velocity must remain fixed.

There are gems at specific points on the race track. Your job is to collect as many gems as possible (they all have the same value).

How many gems can you collect? You may start at any horizontal position you want (but your vertical position must be $0$ at the start).

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line will contain four integers: $n$ ($1 \le n \le 10^5$) is the number of gems, $r$ ($1 \le r \le 10$) is the ratio of vertical velocity to maximum horizontal speed, $w$ ($1 \le w \le 10^9$) is the width of the track, and $h$ ($1 \le h \le 10^9$) is the height of the finish line. Following this will be $n$ lines, each containing an integer $x$ and $y$ coordinate ($0 \le x \le w$, $1 \le y \le h$), containing the coordinate of a gem. All gems will lie on the race track. None will be on the start line.

-----Output-----
Output a single integer on a line by itself representing the maximum number of gems that you can collect.

-----Examples-----
Sample Input 1:
5 1 10 10
8 8
5 1
4 6
4 7
7 9
Sample Output 1:
3

Sample Input 2:
5 1 100 100
27 75
79 77
40 93
62 41
52 45
Sample Output 2:
3
user: There are three horizontal wheels of letters stacked one on top of the other, all with the same number of columns. All wheels have one letter, either ‘A’, ‘B’ or ‘C’, in each of its columns on the edge of the wheel. You may rotate the wheels to adjust the positions of the letters. In a single rotation, you can rotate any single wheel to the right or to the left by one column. The wheels are round, of course, so the first column and last column are adjacent. 

You would like to determine whether it is possible to rotate the wheels so that every column has three distinct letters across the three wheels, and if so, determine the minimum number of rotations required.

-----Input-----
The input has exactly three lines. Each line has a string $s$ ($2 \leq |s| \leq 5 \cdot 10^3$) consisting only of upper-case letters ‘A’, ‘B’ or ‘C’, describing the letters of one wheel in their initial positions. All three strings will be of the same length.

-----Output-----
Output a single integer, which is the minimum number of rotations required, or $-1$ if it isn’t possible.

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

Sample Input 2:
ABBBAAAA
BBBCCCBB
CCCCAAAC
Sample Output 2:
3
user: There are eight planets and one planetoid in the Solar system. It is not a well known fact that there is a secret planet S4 inhabited by small creatures similar to bears, their codename being Lodas. Although this fact is well hidden from the public, the association Savez sent a team lead by general Henrik to study the Lodas. It has been discovered that Lodas have the ability of teleportation and he wants to hire them in his army.

One Lod consists of $N$ strings where the $i$-th string is denoted by $x_ i$. Research has shown that the number of teleportations a Loda can make depends on one special subsequence (not necessarily consecutive) of these strings. Strings $x_ i$ and $x_ j$ ($i < j$) can both be in that sequence if and only if string $x_ j$ both starts with and ends with string $x_ i$. The number of teleportations a Loda can make is the length of the longest described subsequence.

Determine the number of teleportations.

-----Input-----
The first line of input contains of the integer $N$, the number of strings. Each of the following $N$ lines contains one string consisting of uppercase letters of the English alphabet. The input data will be such that there will be less than two million characters in total.

-----Output-----
The first and only line of output must contain the number of teleportations a Loda can make.

-----Examples-----
Sample Input 1:
5
A
B
AA
BBB
AAA
Sample Output 1:
3

Sample Input 2:
5
A
ABA
BBB
ABABA
AAAAAB
Sample Output 2:
3
user: You and your friends are in New York and are planning to go see a broadway musical. Unfortunately, New York being New York, the tickets are just a tiny bit expensive. But one of the shows has a ticket lottery each night where impecunious people such as yourself have a chance to win the right to buy slightly less expensive tickets to good seats. 

The lottery operates as follows. First, everyone interested enters the lottery. Then, $n$ lucky winners are drawn, and each of these is offered to buy up to $t$ tickets.

Given the number of people $p$ in your group (all of which entered the lottery) and the total number of people $m$ that entered the lottery, what is the probability that you will be able to get tickets for your entire group? Assume that the $n$ lucky winners are chosen uniformly at random from the $m$ people that entered the lottery, and that each person can win at most once.

-----Input-----
The input consists of a single line containing four integers:
 - $1 \le m \le 1000$: the total number of people who entered the lottery.
 - $1 \le n \le m$: the total number of winners drawn.
 - $1 \le t \le 100$: the number of tickets each winner is allowed to buy.
 - $1 \le p \le m$: the number of people in your group.

-----Output-----
Output a single line containing the probability that your entire group can get tickets to the show. This probability should be given with an absolute error of at most $10^{-9}$.

-----Examples-----
Sample Input 1:
100 10 2 1
Sample Output 1:
0.1

Sample Input 2:
100 10 2 2
Sample Output 2:
0.1909090909
user: A brokerage firm is interested in detecting automatic trading. They believe that a particular algorithm repeats itself; that is, it makes the same sequence of trades at a later time. The firm has identified a set of 26 key stocks that they believe are likely to be traded in concert, and they’ve encoded a series of trades as a string of letters: the letter itself indicates the stock, upper case indicates a buy, lower case indicates a sell. They want you to write a program to determine, for any two starting points, the longest sequence of identical trades from those two starting points, starting from and including those starting points as the first trade in the sequence.

-----Input-----
There will be a single test case in the input. This test case will start with a string $s$ on the first line, consisting solely of upper case and lower case letters ($1 \le \text {length}(s) \le 100000$). On the next line will be an integer $q$ ($1 \le q \le 100000$), indicating the number of queries. The following $q$ lines each describe a query with two integers, $i$ and $j$ ($0 \le i < j < \text {length}(s)$), which represent two zero-based positions in the string.

-----Output-----
For each query, output a single integer, indicating the length of the longest sequence of trades starting at $i$ which is identical to the sequence of trades starting at $j$.

-----Examples-----
Sample Input:
ABABABcABABAbAbab
3
0 2
1 6
0 7
Sample Output:
4
0
5
user: A cat lives in a tree that has $N$ nodes. She will demarcate her territory by “marking” some of the tree nodes. Marked nodes may not be closer to each other than distance $D$, or the smell will be too overwhelming. Find the maximum number of nodes that the cat can mark.

-----Input-----
First line has two integers, $N$ and $D$. The $0$-th node is the root node of the tree. Then follows $N-1$ lines, the $i$-th of which contain a single integer $x_ i$ with $0 \leq x_ i < i$ (starting with $i = 1$). This means that node $x_ i$ is connected to node $i$.

We always have $1 \leq N, D \leq 2 \cdot 10^5$.

-----Output-----
Output should contain one integer: the maximum number of nodes that can be marked.

-----Examples-----
Sample Input:
4 3
0
0
1
Sample Output:
2
user: There are $n$ cities in the Magical Island, numbered from $1$ to $n$. These cities are connected by $m$ bi-directional roads such that there exists a path between every pair of cities. There may be multiple roads between two cities. Each road has a positive length.

Magical crystals are the most important resources transported between the cities in the Magical Island. The cost of sending one magical crystal from city $s$ to city $t$ equals the smallest distance of any path between city $s$ and city $t$. The distance of a path in the Magical Island is a bit special though: It is the bitwise OR value of all the road lengths on this path.

In each of the next $q$ days, exactly one magical crystal will be transported between a pair of cities. Help the Magical Island figure out the transportation cost for each day.

-----Input-----
The first line contains two integers $n, m$ ($2\leq n\leq 10^5,1\leq m\leq n+200$). Each of the next $m$ lines contains three integers $a, b, w$ that describe a road connecting city $a$ and city $b$ with length $w$ ($1\leq a, b\leq n, a \neq b, 1\leq w \leq 10^9$). In the next line, there is a single integer $q$ $(1\leq q\leq 10000)$, the number of days to consider. The next $q$ lines each have two integers $s$ and $t$ ($1\leq s, t\leq n, s \neq t$). The $i$th line indicates that on day $i$ a crystal is sent from city $s$ to city $t$.

-----Output-----
For each of the $q$ days, output the cost of sending the crystal from $s$ to $t$ for that day.

-----Examples-----
Sample Input:
4 7
1 2 1
1 2 3
1 3 2
1 4 1
2 3 4
2 4 4
3 4 4
3
1 2
1 3
3 4
Sample Output:
1
2
3
user: The Frog Regent has arranged his $N$ frog servants in a circle, with each frog facing the back of the next one. Each frog is assigned a unique integer identifier (ID) from the set of $1$ to $N$. The frog arrangement is specified as a sequence of IDs. The sequence always starts with the frog with the ID $1$. It is followed by the ID of the frog in front of it, then the ID of the next one, and so on until the ID of the last frog – the one behind the frog with ID $1$.

A frog is considered to have made a single leap if it has jumped over the frog in front of it, swapping places with it in the process. For example, if the frogs are sequenced as “1 5 4 3 2 6” and the frog with ID $2$ makes two leaps, the resulting sequence will be “1 2 5 4 3 6” (the frog has shifted two places forward). When the Frog Regent proclaims the number $B$, the frog with ID $B$ makes $B$ leaps.

The Frog Regent wishes, using some number of proclamations, to rearrange the frogs from the starting sequence to his favourite resulting sequence. Given the starting and resulting frog sequences, write a program that will compute a sequence of proclamations needed for the Regent to rearrange the frogs into the resulting sequence. Naturally, the starting and resulting sequences will not be equal.

-----Input-----
The first line of input contains a positive integer $N$, the number of frogs ($3 \le N \le 100$).

The second line of input contains a permutation of the first $N$ positive integers, the starting frog sequence.

The third line of input contains another permutation of the first $N$ positive integers, the resulting frog sequence.

-----Output-----
Output any sequence of integers (one integer per line) that the Frog Regent can proclaim in order to rearrange the frogs into the resulting sequence.

The number of proclamations must not exceed $100000$.

The test data will be such that a solution exists.

-----Examples-----
Sample Input:
6
1 5 4 3 2 6
1 2 5 4 3 6
Sample Output:
2
user: You are hosting a fancy party for fancy friends. And, like any fancy party, you need to buy some fancy antiques to put up around the venue (your house).

There is a set of $n$ fancy antiques that you need to buy. And there is a set of $m$ fancy antique shops in the city. Because these antiques are extremely rare, each fancy antique can only be found at a single fancy antique shop. However, the fancy antique shops can also sell “knock-off” (duplicate) versions of some of the antiques. And of course, for any fancy antique, there is only a single fancy antique shop in the city that holds a knock-off version of that antique (this is to maintain the rareness of the antiques). The shop that sells the original is not always the same shop that holds the knock-off.

It turns out that even though you can tell the difference, most people cannot tell the original version from the knock-off version of any given antique. And, because the shops can get away with it, sometimes the knock-off is more expensive than the original! Since the party is tomorrow, you only have time to visit $k$ shops. You would like to buy one version (either the original or the knock-off) of each of the $n$ antiques.

Suppose that there are three shops, and three antiques we would like to buy.
 - Antique $\# 1$ sells for $30$ at shop $\# 1$. Its knockoff sells for $50$ at shop $\# 2$.
 - Antique $\# 2$ sells for $70$ at shop $\# 2$. Its knockoff sells for $10$ at shop $\# 3$.
 - Antique $\# 3$ sells for $20$ at shop $\# 3$. Its knockoff sells for $80$ at shop $\# 1$.

Suppose you only have time to go to two shops. You can go to shops $1$ and $3$. You can buy the original of antique $1$ with cost $30$ at shop $1$, the original of antique $3$ with cost $20$ at shop $3$, and the knock-off of antique $2$ at shop $3$ with cost $10$. The total cost to buy these items is $60$, which is the minimum possible.

If you only have time to visit one shop, then it is impossible. You cannot buy a version of all three items by visiting a single shop.

Given the costs of the antiques/knock-offs at the shops, what is the minimum total cost to buy one version of each antique?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input will consist of three space-separated integers: $n$, $m$, and $k$ ($1 \le n \le 100$, $1 \le k \le m \le 40$). The next $n$ lines will each have four space separated integers, $a$, $p$, $b$ and $q$, describing an antique, where:
 - $a$ is the index of the shop that sells the original version of the antique ($1 \le a \le m$)
 - $p$ is the price of the original version of the antique at shop $a$ ($1 \le p \le 10^7$)
 - $b$ is the index of the shop that sells the knock-off version of the antique ($1 \le b \le m$)
 - $q$ is the price of the knock-off version of the antique at shop $b$ ($1 \le q \le 10^7$)

-----Output-----
If it is possible to collect all of the antiques while visiting no more than $k$ stores, then output the minimum cost. If it is not possible, output $-1$.

-----Examples-----
Sample Input:
3 3 2
1 30 2 50
2 70 3 10
3 20 1 80
Sample Output:
60
user: You are holding a party. In preparation, you are making a drink by mixing together three different types of fruit juice: Apple, Banana, and Carrot. Let’s name the juices A, B and C.

You want to decide what fraction of the drink should be made from each type of juice, in such a way that the maximum possible number of people attending the party like it.

Each person has a minimum fraction of each of the 3 juices they would like to have in the drink. They will only like the drink if the fraction of each of the 3 juices in the drink is greater or equal to their minimum fraction for that juice.

Determine the maximum number of people that you can satisfy.

-----Input-----
 - One line containing an integer $T$, the number of test cases in the input file.

For each test case, there will be:
 - One line containing the integer $N$, the number of people going to the party.
 - $N$ lines, one for each person, each containing three space-separated numbers “$A$ $B$ $C$”, indicating the minimum fraction of each juice that would like in the drink. $A$, $B$ and $C$ are integers between $0$ and $10000$ inclusive, indicating the fraction in parts-per-ten-thousand. $A + B + C \leq 10000$.

You may assume that $1 \leq T \leq 2$ and $1 \leq N \leq 5000$.

-----Output-----
 - $T$ lines, one for each test case in the order they occur in the input file, each containing the string “Case #$X$: $Y$” where $X$ is the number of the test case, starting from 1, and $Y$ is the maximum number of people who will like your drink.

-----Examples-----
Sample Input:
2
3
10000 0 0
0 10000 0
0 0 10000
3
5000 0 0
0 2000 0
0 0 4000
Sample Output:
Case #1: 1
Case #2: 2
user: You just bought an “artistic” aquarium tank that has an interesting shape, and you poured $L$ litres of water into the tank. How high is the water in the tank?

When you look at this tank from one side, it has the shape of a convex polygon. This polygon has exactly two vertices on the table ($y$-coordinates are 0), and all other vertices have positive $y$-coordinates. There are also exactly two vertices with maximum $y$-coordinates, and water is poured into the opening between these two vertices. This aquarium tank has a depth of $D$ centimetres. The tank is glued to the table, so no matter what shape it has, it keeps its position and does not tip over.

All coordinates and lengths in this problem are given in centimetres. It should be noted that each cubic metre is equivalent to 1 000 litres.

An illustration showing the configuration of the tank of the first sample input is given below:

-----Input-----
The input consists of a single test case. The first line contains an integer $N$ ($4 \leq N \leq 100$) giving the number of vertices in the polygon. The next line contains two integers $D$ and $L$, where $1 \le D \le 1000$ is the depth of the aquarium tank and $0 \le L \le 2000$ is the number of litres of water to pour into the tank. The next $N$ lines each contains two integers, giving the $(x,y)$ coordinates of the vertices of the convex polygon in counterclockwise order. The absolute values of $x$ and $y$ are at most $1000$. You may assume that the tank has a positive capacity, and you never pour more water than the tank can hold.

-----Output-----
Print the height of the water (in centimetres) in the aquarium tank on a line to 2 decimal places.

-----Examples-----
Sample Input:
4
30 50
20 0
100 0
100 40
20 40
Sample Output:
20.83