meetforever/GRPO_CL huggingface dataset:
user: You have arrived in The Big City but your journey is not yet complete. You must still navigate the subway and get to your final destination. The information booth in the subway station is unattended and fresh out of maps of the subway system. On the floor you notice fragments of a map. Can you piece together enough of the map to figure out how to get to your final destination?

Each fragment of the map happens to perfectly contain a single subway station while also identifying all of the other stations that it connects to. Each connection between stations is bi-directional such that it can be travelled going either direction. Using all of the available fragments, your task is to determine the sequence of stations you must pass through in order to reach your final destination or state that there is no route if you don’t have enough information to complete your journey.

-----Input-----
The first line of input has an integer, $2 \leq N \leq 32$, that identifies the number of pieces of the map that were found.

The following $N$ lines each describe a station depicted on one of those pieces. Each of these lines starts with the name of the station they describe and is followed by a space-separated list of all of the station names that are directly connected to that station (there may be as many as $N-1$).

The final line identifies a starting station and a destination station. The destination station is guaranteed to be different than the starting station.

Each station name is a string of up to $20$ characters using only letters a–z and A–Z. It is guaranteed that there is at most one simple route (without revisiting stations) from the starting station to the destination station.

-----Output-----
Give the sequence of stations that leads from the starting station to the destination station. Separate station names with spaces. If there are not enough pieces of the map to find a route from the starting station to the destination station then output “no route found”.

-----Examples-----
Sample Input 1:
3
Uptown Midtown
Midtown Uptown Downtown
Downtown Midtown
Uptown Downtown
Sample Output 1:
Uptown Midtown Downtown

Sample Input 2:
6
A B
B A D
C D
E D F G
F E
G E
F A
Sample Output 2:
F E D B A
user: Being a responsible young adult, you have decided to start planning for retirement. Doing some back-of-the-envelope calculations, you figured out you need at least $M$ euros to retire comfortably.

You are currently broke, but fortunately a generous gazillionaire friend has offered to lend you an arbitrary amount of money (as much as you need), without interest, to invest in the stock market. After making some profit you will then return the original sum to your friend, leaving you with the remainder.

Available to you are $n$ investment opportunities, the $i$-th of which costs $c_ i$ euros. You also used your computer science skills to predict that the $i$-th investment will earn you $p_ i$ euros per day. What is the minimum number of days you need before you can pay back your friend and retire? You can only invest once in each investment opportunity, but you can invest in as many different investment opportunities as you like.

For example, consider the first sample. If you buy only the second investment (which costs $15$ euros) you will earn $p_2 = 10$ euros per day. After two days you will have earned $20$ euros, exactly enough to pay off your friend (from whom you borrowed $15$ euros) and retire with the remaining profit ($5$ euros). There is no way to make a net amount of $5$ euros in a single day, so two days is the fastest possible.

-----Input-----
 - The first line contains the number of investment options $1 \leq n \leq 10^5$ and the minimum amount of money you need to retire $1 \leq M \leq 10^9$.
 - Then, $n$ lines follow. Each line $i$ has two integers: the daily profit of this investment ${1 \leq p_ i \leq 10^9}$ and its initial cost $1 \leq c_ i \leq 10^9$.

-----Output-----
Print the minimum number of days needed to recoup your investments and retire with at least $M$ euros, if you follow an optimal investment strategy.

-----Examples-----
Sample Input 1:
2 5
4 10
10 15
Sample Output 1:
2

Sample Input 2:
4 10
1 8
3 12
4 17
10 100
Sample Output 2:
6
user: A regular bracket-sequence is a string of characters consisting only of opening and closing brackets, and satisfying the following conditions:
 - An empty string is a regular bracket-sequence.
 - If $A$ is a regular bracket-sequence, then ($A$), [$A$] and {$A$} are also regular bracket-sequences.
 - If $A$ and $B$ are regular bracket-sequences, then $AB$ is also a regular bracket-sequence.

For example, the sequences “[({})]”, “[](){}” and “[{}]()[{}]” are regular, but the sequences “[({{([”, “[]({)}” and “[{}])([{}]” are not.

Ivica has found a string which looks like it could be a regular bracket-sequence. Some of the characters have become smudged and illegible, and could have been any character.

Write a program that calculates how many ways the illegible characters in the string can be replaced by brackets so that the result is a regular bracket-sequence. This number can be very large, so output only its last $5$ digits.

-----Input-----
The first line contains an even integer $N$ ($2 \le N \le 200$), the length of the string. The second line contains the string. Illegible characters are represented by the ‘?’ character.

-----Output-----
Output the number of regular bracket-sequences the string could have read.

-----Examples-----
Sample Input 1:
6
()()()
Sample Output 1:
1

Sample Input 2:
10
(?([?)]?}?
Sample Output 2:
3
user: At one of the many climate protests, Skylar fell in love with a fellow environmental activist. Unfortunately, the two young lovers live on opposite sides of the planet and long distance travel is only practical by (gasp) air. Skylar had scolded friends and family for flying, heavily handing out the recent Swedish export flygskam (verbatim translation: flight shame). But alas, the things we do for love! Now they want your help to calculate the minimum amount of flygskam Skylar will accumulate on a one-way trip across the globe. 

To calculate the best route you models the planet as a perfect sphere and assumes that all flights fly at the distance $6381$ km from the center of the earth. The amount of shame for a single point-to-point flight is calculated as the distance between the airports in km, plus a take-off and landing penalty of $100$, that is, two airports with the flight distance $1000$ km will result in $1100$ shame.

-----Latitude and longitude-----
The positions of the airports are given as the latitude and longitude in (decimal) degrees. The latitude of a point $P$ on the earths surface is the angle between the equatorial plane and a line passing through $P$ and the center of the earth. The equator has latitude $0^\circ $, points north of the equator has positive values and points south of the equator has negative values, the North Pole has latitude $90^\circ $ and the South Pole latitude $-90 ^\circ $. Half circles that run from the North to the South pole are called meridians. The zero meridian runs through Greenwich. The longitude of a point $Q$ is the angle between the zero meridian plane and the line that run through $Q$ and the center of the earth, with values from $- 180^\circ $ to $+180^\circ $, with positive values east of Greenwich.

-----Input-----
Input starts with one line with two integers $1 \leq N \leq 10000$, the number of airports and $1 \leq M \leq 100000$, the number of two-way flight routes. The second line has two strings $S$ and $T$, Skylar’s start position and Skylar’s target position. Then follows $N$ lines, each starts with a three letter (uppercase) airport code, followed by two real values numbers, the latitude and longitude in degrees. Then follows $M$ lines, each with two strings $a$ and $b$, the airports with a two-way flight connection.

All flight airports have unique names and all connections are between existing airports.

-----Output-----
Output a real value with the minimum amount of flygskam Skylar will obtain on a one-way trip. If the target is unreachable and Skylar will be forever alone, output -1. Answers within a relative or absolute error of $10^{-6}$ will be accepted.

-----Examples-----
Sample Input 1:
4 4
ARN SCR
ARN 59.6467921 17.9370443
SCR 61.156603 12.837360
CPH 55.618023 12.650763
OSL 60.197646 11.100008
ARN OSL
ARN CPH
SCR OSL
OSL CPH
Sample Output 1:
729.09706162045

Sample Input 2:
2 1
LAX AKL
AKL -37.006131 174.783781
LAX 33.941589 -118.408531
LAX AKL
Sample Output 2:
10603.3297338597
user: You are a camp counselor at a summer camp and it is time to take some of the kids to lunch. There are $n$ groups of friends of different sizes who would like to be able to eat lunch together. There are $k$ bins that each hold exactly $x$ lunches. If one of the lunches in a bin is not used, then it must be thrown out due to health regulations.

If you take fewer than $a$ students to lunch then your fellow counselors will get angry as they have to watch the remaining campers. However, due to child safety regulations, you can watch at most $b$ children at any given time.

Is it possible to take a subset of the students to lunch such that the children get to eat with their friends, there are no wasted lunches, you take no more than the legal limit of campers, and you don’t anger your fellow counselors by taking too few?

-----Input-----
The first line contains one integer, $n$ ($1 \leq n \leq 1000$), the number of groups of friends. The next line contains $n$ space separated integers, $s_0, s_1, \ldots , s_{n-1}\ (0 < s_ i \leq 100)$, where $s_ i$ denotes the size of group $i$. The next line contains $4$ space separated integers: $k$ ($0 < k \leq 100$), $x$ ($0 < x \leq 100000$), $a$ and $b$ ($0 < a \leq b \leq 10000$) representing the number of bins, the number of lunches in each bin, the minimum number of campers you must watch, and the maximum number of campers you are allowed to watch, respectively.

-----Output-----
Output an integer denoting the maximum number of students you can take to lunch and satisfy the requirements, or, if it is not possible, output ‘impossible’.

-----Examples-----
Sample Input 1:
7
5 7 10 15 3 2 8
20 3 30 40
Sample Output 1:
39

Sample Input 2:
7
33 7 10 15 3 2 8
20 3 30 40
Sample Output 2:
36
user: Mirko and Slavko’s favourite pastime is competing against each other in mathematical games. This time they took a heap of $N$ pebbles and settled on the following rules:
 - Mirko is the first to play, then Slavko, then Mirko again, then Slavko and so on;
 - Mirko can take any number of pebbles (between $1$ and $N$, inclusive) from the heap during his first move;
 - In each of the following turns the current player must take at least $1$ pebble and is allowed to take at most double the amount of pebbles taken during the previous turn by the other player; naturally, one cannot take more pebbles than the remaining amount in the heap;
 - The player who takes the last pebble is the winner.

Both Mirko and Slavko play optimally (if it is possible for one player to beat the other, that player will always win). We need to find the minimum number of pebbles that Mirko must take during his first turn such that he is guaranteed to win the game.

-----Input-----
The first and only line of input contains the positive integer $N$ ($2 \le N \le 10^{15}$), the number of pebbles in the starting heap.

-----Output-----
The first and only line of output must contain the required minimum number of pebbles that Mirko needs to remove during his first turn.

-----Explanation of Sample Input 1-----
Mirko has $4$ possibilities to choose from: he can take $1$, $2$, $3$, or $4$ pebbles from the heap. If he takes all $4$ pebbles he will naturally win, but that is not the minimum solution. We need to check the remaining alternatives. If Mirko takes only one pebble, Slavko is left with a heap of $3$, but he can take at most $2$. Slavko cannot take all pebbles, but Mirko will be able to take all remaining pebbles during his next turn, winning the game. We conclude that $1$ is the minimum solution for this test case.

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

Sample Input 2:
7
Sample Output 2:
2
user: Gerald’s job is to welcome the teams for this year’s NWERC at the airport in Linköping. One of his duties is to stand at the luggage carousel and collect all the knapsacks that the teams are bringing. Gerald is a lazy person, so he just stands at the same position of the carousel and waits for bags to pass by so he can pick them up. 

The baggage carousel consists of $s$ luggage slots, numbered in ascending order from $0$ to $s-1$. Since the baggage carousel is cyclic, luggage slots $s-1$ and $0$ also lie side by side. The carousel turns in such a way that if Gerald stands in front of slot $i$ at some point in time, he will stand in front of slot $(i+1) \bmod s$ one time unit later.

In the beginning Gerald prepares a huge baggage cart at some position and stands there to wait for luggage. When a knapsack arrives in front of Gerald, he needs $t$ time units to take it and put it on the baggage cart. After these $t$ time units he is ready to pick up another knapsack. As long as there are remaining knapsacks on the luggage carousel, Gerald always takes the next one to arrive at his position as soon as he is ready after putting away the previous one.

Now Gerald wonders about the effect of his choice of position on the time it will take him to finish this task. It is up to you to help Gerald calculate the minimum, maximum, and average time to pick up all knapsacks, taken over all $s$ possible slots, which can appear in front of Gerald after preparation. Time starts when he has prepared the baggage cart at some slot of the baggage carousel and ends after he has put the last knapsack on the cart.

-----Input-----
The input consists of:
 - one line with three integers $n$ ($1\le n\le 2000$), $s$ ($1\le s\le 10^7$) and $t$ ($1\le t \le 10^7$), where $n$ is the number of knapsacks to pick up, $s$ is the number of slots of the carousel, and $t$ is the number of time units Gerald needs to pick up a knapsack from the carousel and put it on the cart;
 - one line with $n$ integers $k_1, \ldots , k_ n$ ($0 \le k_ i \le s-1$ for $1 \le i \le n$), the slots of the knapsacks.

There may be several knapsacks stacked on top of each other in the same slot, but Gerald can still only pick up one knapsack at a time.

-----Output-----
Output three lines of output containing the minimum, maximum, and average time to pick up all the luggage, over all $s$ positions. The average time should be output as a reduced fraction in the form $p/q$.

-----Examples-----
Sample Input 1:
7 10 10000000
0 0 0 0 0 0 1
Sample Output 1:
70000001
70000009
350000027/5

Sample Input 2:
10 10 3
0 0 2 2 4 4 6 6 8 8
Sample Output 2:
39
40
79/2
user: In C++ division with positive integers always rounds down. Because of this, sometimes when two integers are divided by the same divisor they become equal even though they were originally not equal. For example in C++, $\verb|5/4|$ and $\verb|7/4|$ are both equal to 1, but $5 \neq 7$.

Given a list of nonnegative integers and a divisor, how many pairs of distinct entries in the list are there that give the same result when both are divided by the divisor in C++?

-----Input-----
The first line of input contains two integers $n$ ($1 \leq n \leq 200000$), the number of elements in the list, and $d$ ($1 \leq d \leq 10^9$), the divisor.

The second line of input contains $n$ integers $a_1, \ldots , a_ n$ ($0 \leq a_ i \leq 10^9$), where $a_ i$ is the $i^\textrm {th}$ element of the list.

-----Output-----
Display a single integer indicating the number of distinct pairs of indices $(i,j)$ with $1 \leq i < j \leq n$ such that $a_ i / d = a_ j / d$ when using integer division in C++. Note that the numbers in the list are not necessarily distinct (i.e. it is possible that $a_ i = a_ j$ for some indices $i \neq j$).

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

Sample Input 2:
5 1
4 5 6 7 8
Sample Output 2:
0
user: Han didn’t want to study solo so he invited his friend Dominik to come over. After an eventful evening that will be remembered for a record number of solved tasks from the field of electronics, Dominik went home. To his surprise, the police stopped him thinking he was drunk. It is known that in these situations sobriety is proven by solving a series of carefully crafted tasks that test a man’s cognitive abilities. If we can trust Dominik, the conversation went something like this:Policeman

Something easy to begin with… What is the complexity of bubble sort?Dominik

That is really easy, $\mathrm{O}(n^2)$.Policeman

Say the English alphabet in reverse.Dominik

Trivial, zyxwvutsrqponmlkjihgfedcba.Policeman

You learned that by heart. Now imagine that all the letters of the English alphabet from ‘a’ to ‘z’ are respectively written clockwise in a circle. Begin with the letter ‘a’ and say the letters clockwise. After each spoken letter, I can tell you to continue saying the alphabet in reverse order or I can ask you how many times so far you’ve said a certain letter. Are you ready? 3, 2, 1, Go!Dominik

Um… a, b, c…

Write a programme that solves Dominik’s problem.

-----Input-----
The first line of input contains the integer $Q$ ($1 \leq Q \leq 100000$), the number of policeman’s orders. Each of the following $Q$ lines contains one of the policeman’s order in the form of “SMJER $n$” (Croatian for direction) or “UPIT$n$ $x$” (Croatian for query). The order in the form “SMJER $n$” means that, after the nth spoken letter, Dominik must start saying the alphabet in reverse, whereas the order in the form “UPIT$n$ $x$” means that Dominik must say how many times so far he’s said the letter $x$ in the first $n$ spoken letters.

The policeman’s order will be given chronologically in the input, or, the numbers $n$ ($1 \leq n \leq 10^9 $) from the orders will be strictly ascending. The character $x$ from the order in the form of “UPIT $n$ $x$” is a lowercase letter of the English alphabet.

-----Output-----
For each order in the form of “UPIT $n$$x$”, output how many times Dominik has said the letter $x$ in the first $n$ spoken letters. The answer to each query needs to be written in a separate line, and the queries need to be answered in the order given in the input.

-----Examples-----
Sample Input 1:
5
UPIT 1 b
UPIT 3 b
SMJER 4
UPIT 7 a
UPIT 10 z
Sample Output 1:
0
1
2
1

Sample Input 2:
5
SMJER 1
SMJER 2
SMJER 3
UPIT 5 a
UPIT 7 w
Sample Output 2:
2
1
user: Given a directed weighted graph and two vertices $s$, $t$, the goal is to find a subset $U$ of the vertices such that $s \in U$, $t \not\in U$, and the weight of edges from $U$ to $\overline{U}$ is minimized.

-----Input-----
The first line of input contains four non-negative integers, $2 \le n \le 500$, $0 \le m \le 10000$, $0 \le s \le n-1$ and $0 \le t \le n-1$, separated by single spaces, where $n$ is the numbers of nodes in the graph, $m$ is the number of edges, $s$ is the source and $t$ is the sink ($s \ne t$). Nodes are numbered from $0$ to $n-1$. Then follow $m$ lines, each line consisting of three (space-separated) integers $u$, $v$ and $w$ indicating that there is an edge from $u$ to $v$ in the graph with weight $1 \le w \le 10^8$.

-----Output-----
Output should begin with a line containing an integer $k$, giving the size of $U$. Then follow $k$ lines giving the vertices in $U$, one per line. If there are multiple choices for $U$ any one will be accepted.

You may assume that there is a cut such that the total weight of edges from $U$ to $\overline{U}$ is less than $2^{31}$.

-----Examples-----
Sample Input 1:
4 5 0 3
0 1 10
1 2 1
1 3 1
0 2 1
2 3 10
Sample Output 1:
2
1
0

Sample Input 2:
2 1 0 1
0 1 100000
Sample Output 2:
1
0
user: It’s another day in the office, and you’re a mastermind of not doing any work yourself. Instead, you’ll go to your coworkers for “help,” but secretly have them do all the work.

You’ve determined that the more one of your coworkers helps you, the more annoyed they become. You’ve also been able to determine how much more annoyed a coworker gets everytime you ask them for help. At the beginning of the day, a coworker is initially $a$ annoyed at you. That’s their annoyance level. Everytime you ask them for help though, they become $d$ more annoyed at you – their annoyance level $a$ increases by a constant amount $d$ so that $a=a+d$.

You want to complete a project of $h$ tasks solely with “help” from your coworkers, but you need to be careful not to annoy any of them too much.

What’s the best you can do?

-----Input-----
The first line contains $2$ integers $h$ and $c$, where $h$ ($1 \le h \le 100000$) is the number of times you have to ask for help to complete the project, and $c$ ($1 \le c \le 100000$) denotes the number of coworkers you have.

Each of the following $c$ lines contains two positive integers $a$ and $d$, representing a coworker whose initial annoyance level is $a$ and who is getting more annoyed at you by an increase of $d$ every time you ask them for help ($1\le a, d \le 10^9$).

-----Output-----
Output a single number, which is the maximum annoyance level any coworker has at you provided you use an optimal strategy to minimize this level. (In other words, of all possible strategies, choose one that minimizes the annoyance level of the worker or workers who are most annoyed at you at the end.)

-----Sample Input 1 Explanation-----
You have $4$ coworkers and you need to ask for help $4$ times. Initially, their annoyance levels are $a_1=1, a_2=2, a_3=3, a_4=4$, the increases are $d_1=2, d_2=3, d_3=4, d_4=5$. One optimal solution is to ask for help twice from coworker $1$, once from coworker $2$, and once from coworker $3$, in which case the final annoyance levels are: $a_1=1 + 2 \cdot 2 = 5, a_2=2 + 3 = 5, a_3=3 + 4 = 7, a_4=4$. The coworker that is most annoyed at you is coworker $3$, whose annoyance level at you is $7$. Or, you could ask coworker $1$ for help $3$ times and coworker $2$ once, leaving you with $a_1=1 + 3 \cdot 2 = 7, a_2=2 + 3 = 5, a_3=3, a_4=4$. Both strategies yield the same minimal maximum amount.

-----Examples-----
Sample Input 1:
4 4
1 2
2 3
3 4
4 5
Sample Output 1:
7

Sample Input 2:
3 2
1 1000
1000 1
Sample Output 2:
1002
user: In a rainforest there are $n$ treehouses high in the forest canopy on different trees (numbered from $1$ to $n$). The $i$-th tree’s location is at $(x_ i, y_ i)$. The first $e$ of them in the list are close enough to neighboring open land around the rainforest so that transportation between all of them is easy by foot. Some treehouses may already be connected by direct straight cables through the air that can allow transport between them.

Residents want easy transportation between all the treehouses and the open land, by some combination of walking (between those near the open land), and using one or more cables between treehouses. This may require the addition of more cables. Since the cables are expensive, they would like to add the smallest possible length of cable.

The height of a cable up two trees can be set so cables can criss-cross other cables, and not allow any snags or crashes. It is not safe to try to switch between two criss-crossed cables in mid-air!

-----Input-----
The input will start with the three integers $n$ ($1 \le n \le 1000$), $e$ ($1 \le e \le n$), and $p$ ($0 \le p \le 1000$), where $p$ is the number of cables in place already.

Next come $n$ lines, each with two real numbers $x$ and $y$ ($|x|, |y| \le 10000$) giving the location of a treehouse. The $i$-th coordinate pair is for the treehouse with ID $i$. All coordinate pairs are unique. Real numbers are stated as integers or include one digit after a decimal point.

Next come $p$ lines, each with two integers $a$, $b$, where $1 \le a < b \le n$, giving the two treehouse ids of an existing cable between their trees. No ID pair will be repeated.

-----Output-----
The output is the minimum total length of new cable that achieves the connection goal, expressed with absolute or relative error less than $0.001$.

-----Examples-----
Sample Input 1:
3 1 0
0.0 0.0
2.0 0.0
1.0 2.0
Sample Output 1:
4.236067

Sample Input 2:
3 1 1
0.0 0.0
0.5 2.0
2.5 2.0
1 2
Sample Output 2:
2.000000