meetforever/GRPO_CL huggingface dataset:
user: Mr. G is one of the most excellent students in North River High School for Gifted Students. Despite having impressive performance in a programming competition and making it to the next round, he was not totally happy since his best friend did not get such a great achievement. In order to appease the poor girl, Mr. G has to deal with a very hard challenge: Buy her some ginger candies!

The road system in North River province consists of $N$ junctions and $M$ bidirectional roads. Junctions are numbered from $1$ to $N$, and roads are numbered from $1$ to $M$, inclusive. On the $i^{th}$ road, which connects two junctions $u_ i$ and $v_ i$, there is a shop in which Mr. G can buy $c_ i$ ginger candies. No two roads have the same number of candies. Mr. G wants to meet his friend at some junction, travel through several roads without visiting the same road twice, buy all candies on those roads, and finish at the same junction where he starts.

Using his humble knowledge in Physics, Mr. G calculates the amount of energy he needs to spend as follow: Let $L$ be the maximum number of candies he buys in one road, and $K$ be the number of roads he passes through. The amount of energy he needs to spend is $L^2+\alpha K$, where $\alpha $ is some constant he has already known.

Help him to satisfy his friend with the minimum amount of energy.

-----Input-----
 - The first line contains three integers $N$, $M$, $\alpha $, the number of junctions, the number of roads and the predefined constant Mr. G uses to calculate the amount of energy, respectively ($1 \leq N \leq 10^5$, $1 \leq M \leq 2 \times 10^5$, $1 \leq \alpha \leq 20$).
 - In the next $M$ lines, each contains three integers $u$, $v$, $c$ ($1 \leq u \leq N$, $1 \leq v \leq N$, $10^6 \leq c \leq 10^9$), meaning that there is a road connecting two junctions $u$ and $v$, which sells $c$ ginger candies.

It is guaranteed that all $c$ in the above $M$ lines are distinct.

-----Output-----
Write one integer denoting the minimum amount of energy Mr. G has to spend. If there is no route satisfying the condition, output Poor girl instead.

-----Examples-----
Sample Input 1:
7 7 10
1 2 1000000
2 3 2000000
3 4 3000000
4 5 4000000
5 6 5000000
6 7 6000000
7 1 7000000
Sample Output 1:
49000000000070

Sample Input 2:
6 6 7
1 3 1000000
3 5 3000000
5 1 5000000
2 4 2000000
4 6 4000000
6 2 6000000
Sample Output 2:
25000000000021
user: Emil loves his electric moped. He loves it so much that he rides it through rain, snow, wind and ice. Some day, he will be able to fly with it. This weekend, Emil is on a vacation in New York. He has compiled a list of cool places to visit, as well as a list of all one-way streets between these places, and now it is time for some sightseeing! 

One sightseeing trip consists of $S$ places and exactly $S$ streets which connect them. He starts by travelling to one of these places by subway, with his moped on his back. Then he starts riding his moped to each place of the trip in turn, finally returning to where he started in the current trip. Then he goes back to the subway, with his moped on his back, and goes for the next trip. He does this until he has enjoyed all of the trips.

Since it’s very hard to determine if Emil can visit all of the places he wants to see in a single trip of this kind, he has instead asked you to help him split all of the $N$ places in town into a number of trips, such that each place in town is included in exactly one trip.

Given all the places Emil wants to see and the streets of the city connecting these, construct a list of trips fulfilling these conditions, or determine that it is impossible. Each trip must use at least one street, and there may be loops, i.e. streets with a start and end point in the same place.

-----Input-----
The first line of input consists of two integers $1\le N \le 100$ and $0 \le M \le 10000$, the number of places Emil wants to visit and the number of roads respectively. The places are numbered $0$ to $N-1$.

The next $M$ lines consists of two integers $0 \le f, t \le N-1$, denoting that there is a one-way street from place $f$ to place $t$.

-----Output-----
Output begins with an integer $N$ on its own line, the number of trips made. Then, for every trip, output a line with the number of places included in the trip $K$, followed by $K$ lines, containing the numbers of said places in order of visit (the last place which Emil returns to should not be output twice).

If the places can’t be split into trips, Emil instead goes bowling. Then just output the string "Yeah!". If there are several solutions, output any of them.

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

Sample Input 2:
4 4
0 1
1 0
2 3
3 3
Sample Output 2:
Yeah!
user: Anton, a young mathematician, comes from the cold region that is northern Sweden. He likes it there, since there are a lot of trees. Lately, however, this has made him a bit sad, since he can’t see the forest because of all the trees. Therefore, he has asked you to construct a forest.

Since Anton is a mathematician, he doesn’t want just any forest – he wants a graph theoretical forest. A forest is a (possibly disconnected) graph without cycles – i.e., a union of disjoint trees.

Anton has already decided how many vertices he wants in his forest, as well as the degree of each vertex. Now, it is your task to construct a forest according to his wishes, or determine that no such forest exists.

-----Input-----
The first line of the input is a single integer $0 \le V \le 100$, the number of vertices in his forest.

The next line contains $V$ integers $d_1, d_2, \ldots , d_ V$ ($0 \le d_ i \le 100$ for all $i$). These are the degrees which Anton wants the vertices to have.

-----Output-----
The first line of output should be IMPOSSIBLE if it is impossible to construct such a forest.

Otherwise, the first line should be POSSIBLE. The next lines should then contain two integers $1 \le a, b \le V$ each, denoting an edge between vertices $a$ and $b$. Note that the vertex with number $a$ must have exactly degree $d_ a$, i.e. the ordering of the vertices is important.

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

Sample Input 2:
2
1 2
Sample Output 2:
IMPOSSIBLE
user: You are given four points in space (i.e. $\mathbb {R}^3$). The points are not coplanar. In other words, there is no plane in space that contains all four points.

It is guaranteed that under such assumption, there exists a (unique) sphere whose surface touches all four points. Find the coordinates of the centre of the sphere.

-----Input-----
The input consists of four lines. Line $i$ consists of three integers $x_ i$, $y_ i$, $z_ i$, representing the coordinates $(x_ i, y_ i, z_ i)$ of point $i$.

For all test cases, $-100 \leq x_ i, y_ i, z_ i \leq 100$.

-----Output-----
Output three numbers (not necessarily integers) $x_ c$, $y_ c$, $z_ c$, representing the coordinates $(x_ c, y_ c, z_ c)$ of the centre of the sphere that touches all four points.

Let the jury’s answer be $(x_ j, y_ j, z_ j)$. Your answer will be considered as correct if and only if\[ \sqrt {(x_ j - x_ c)^2 + (y_ j - y_ c)^2 + (z_ j - z_ c)^2} < 10^{-4}. \]

-----Examples-----
Sample Input 1:
0 0 0
1 0 0
0 1 0
0 0 1
Sample Output 1:
0.5 0.5 0.5

Sample Input 2:
-1 0 0
1 0 0
0 1 0
0 0 1
Sample Output 2:
0.0 0.0 -0.0
user: A very popular position has just opened up at Poodle Inc. Candidates have formed a queue while they wait for their turn to be interviewed.

Competition is fierce and each candidate knows that they will not be selected if another candidate is strictly better than them. Every minute, each candidate looks at the resumé of candidates who are currently adjacent to them in the queue (ahead and behind). If at least one of the neighbour’s resumé’s perceived value is strictly greater than their resumé’s perceived value, they leave the queue since they do not want to waste their time. The candidates all look at their neighbor’s resumé simultaneously, and then some candidates leave the queue simultaneously.

This process continues until no more candidates leave the queue. Determine the number of minutes that pass while this process takes place. Report the values of the candidates that leave the queue in each round. Also report the final state of the queue.

-----Input-----
The first line of input contains a single integer $N$ ($1 \leq N \leq 100000$), which is the number of candidates.

The second line contains $N$ integers $v_1, \ldots , v_ N$ ($0 \leq v_ i \leq 10^9$ for each $1 \leq i \leq N$), where $v_ i$ is the perceived value of the $i^\textrm {th}$ candidate.

-----Output-----
Display $M$, the number of minutes taken by this process. Then display $M$ lines. The $i^\textrm {th}$ line should contain the perceived values of the candidates who left the queue in the $i^\textrm {th}$ minute in the same relative order that they appeared in the queue. Finally display a line indicating the final list of perceived values in the queue after candidates no longer leave it.

-----Examples-----
Sample Input 1:
10
3 6 2 3 2 2 2 1 5 6
Sample Output 1:
2
3 2 2 1 5
3 2 2
6 6

Sample Input 2:
3
17 17 17
Sample Output 2:
0
17 17 17
user: In Croatia there are $N$ cities connected by $E$ two-way roads. Two large food chains have recently reached an agreement on market sharing. In the middle of each road, exactly one chain will be given rights to build a restaurant.

To ensure the market is shared fairly, each city must have at least one restaurant from each chain on the roads connected to that city. However, there are cities with only one road, or no roads at all, and for those cities the above requirement does not apply. Such cities are doomed to visit one chain, or travel a bit further.

Write a program that will determine for each road the chain that should build there so that these requirements are met.

-----Input-----
The first line of input contains two integers $N$ and $E$ ($1 \le N, E \le 100000$), number of cities and number of roads.

The next $E$ lines contain two integers each. Each line describes one road. Integers $A_ i$ and $B_ i$ ($1 \le A_ i , B_ i \le N$; $A_ i \ne B_ i$) denote a road connecting cities $A i$ and $B_ i$.

There will never be two or more roads connecting the same pair of cities.

-----Output-----
If there is no way to fairly assign the roads, the first and only line of input should contain “0”.

Otherwise output exactly $E$ lines, one for each road, in the same order as they were given in the input. The $i^\text {th}$ line should contain “1” if the first chain has the right to build on this road, or “2” if the second one does.

Note: if the solution is not unique, you may output any valid solution.

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

Sample Input 2:
7 7
1 2
2 3
3 1
4 5
5 6
6 7
7 4
Sample Output 2:
0
user: The most adventurous part of skiing is the journey onto the mountain-top, between trees and through clouds, and past all sorts of enchanting views.

Naturally, the skiers at the foot of the lift can hardly wait to take their turns (although they are a little disappointed that the climb will eventually terminate). They all know exactly which times they plan to catch a lift on the tireless rotary machine.

Unfortunately, there are only so many gondolas available at the start of the day to attach to the track loop. The track loop takes $2 \cdot T$ minutes to cycle around ($T$ on the way up and then $T$ on the way back down). Given that you can arrange the gondolas on the track however you see fit, what is the minimum summed waiting time of all skiers that you can achieve?

-----Input-----
 - One line containing three integers:
 - $N$ ($1 \le N \le 400$), the number of skiers.
 - $T$ ($1 \le T \le 720$), the time to travel from the bottom of the hill to the top in minutes.
 - $G$ ($1 \le G \le 400$), the number of available gondola cabs.
 - A further $N$ lines in no particular order, each containing one integer $X$ ($0 \le X \le 10^6$) which gives the time when a skier arrives at the foot of the mountain.

-----Output-----
 - One line containing one integer: the minimum possible sum of all waiting times, where the waiting time for one skier is the time difference between their arrival and departure on the next available gondola (which may be shared with any number of other skiers).

-----Examples-----
Sample Input 1:
4 10 2
0
15
30
45
Sample Output 1:
10

Sample Input 2:
4 10 3
0
15
30
45
Sample Output 2:
5
user: When it comes to trams, a lot of people are civilized individuals who know how to behave in one. However, there are always those few who upon spotting a place to sit will run towards it in supersonic speeds. If they spot more than one place, they always try the closest one first.

Problems arise when two or more such individuals aim for the same spot. If one of them is the closest, he or she will sit, and others won’t even attempt to move in and instead turn their attention to the next closest spot. If however they are all equally close, they will all run to the seat resulting in a massive explosion that usually ends with complete destruction of both them and the seat.

You are given a description of one tram. It is represented as a table with $R$ rows and $C$ columns. The rude passengers are marked with the letter ‘X’. Empty seats are marked with ‘L’ and the tram floor is marked with ‘.’. Note that although there are other passengers, the force projected by these idiots is more than enough to simply walk through them.

Distance between two cells is the Euclid distance between their centers. Write a program that will determine the number of explosions which will take place before all people are seated, or destroyed, or they run out of chairs.

-----Input-----
The first line of input contains two integers, $R$ $(1 \leq R \leq 100)$ and $C$ $(1 \leq C \leq 100)$, number of rows and columns.

The next $R$ lines contain $C$ characters each. ‘.’, ‘X’ or ‘L’. There will always be at least one character ‘X’ and at least one ‘L’ in the input. Also, there will be no two ‘L’ characters such that they are both equally distant to some ‘X’.

-----Output-----
The first and only line of input should contain the number of explosion for the given layout.

-----Examples-----
Sample Input 1:
4 4
.LX.
.X..
....
.L..
Sample Output 1:
1

Sample Input 2:
4 4
.XLX
.X..
...L
.X..
Sample Output 2:
2
user: High school is all about being in the coolest group of friends. Headmistress Umbridge knows this, and she also knows that knowledge is power. She has collected data on all of the $n$ students at the school, asking each of them who they are friends with. Now she has a list of responses, but she is suspicious that some of the students might not have been entirely truthful during the questioning.

From anonymous (but highly reliable) sources, Headmistress Umbridge knows that the friendships at her school satisfy the following properties:
 - If $a$ is friends with $b$ then $b$ is also friends with $a$.
 - The set of students can be partitioned into groups, such that every student participates in exactly one group, where
 - each group has at least one and at most $p$ students, and
 - for each group there are at most $q$ pairs of friends with the first one in the group, and the second one outside of it.

Note that two students in the same group do not have to be friends.

Umbridge has hired you to figure out whether it is possible that all students are telling the truth, or whether she can be sure that at least one student is lying, and that she therefore should put everyone in detention. Is this morally questionable? Probably.

(In case the students may be telling the truth, you are worried that her suspicion might fall on you instead; thus you will also want to provide evidence of a valid partition if there is one.)

-----Input-----
First a single line with three non-negative integers $n$, $p$ and $q$ as described above. Next follow $n$ lines, one for each student, starting with student $i=0$. Each such line starts with an integer $m_ i$, denoting the number of friends student number $i$ claims that she has. Then follow $m_ i$ distinct integers between $0$ and $n-1$, indicating who those friends are (the students are numbered from $0$ to $n-1$).

We always have $1 \leq n \leq 2500$, and $p + q \leq 15$. Furthermore, it is guaranteed that $m_0 + m_1 + \ldots + m_{n-1} \leq 30000$. A student never lists herself as one of her friends.

-----Output-----
If Dolores can be certain someone did’t tell the truth, output “detention”. Otherwise, output “home”. If you output home on the first line, then you should prove your claim by outputting a partition of the students into groups such that the requirements above hold (if there are several, any one will do): The second line should then contain a positive integer $G$, the number of groups. The following $G$ lines should each begin with a positive integer $g_ i$, the number of students in the $i$-th group. Then on the same line, $g_ i$ integers indicating the students in this group.

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

Sample Input 2:
5 2 1
1 1
2 0 2
2 1 3
2 2 4
1 3
Sample Output 2:
detention
user: Consider a string of $N$ letters, where $N$ is an even number. Rearrange the letters such that all $N/2+1$ substrings of length $N/2$ are different.

-----Input-----
A single line containing string $S$ of length $N$, where $2 \leq N \leq 10^5$ and $N$ is even. The string consists of lower-case letters from the English alphabet, “a” to “z”.

-----Output-----
If possible, print a string of length $N$ that is a rearrangement of the letters in $S$ such that all substrings of length $N/2$ are different. If this is impossible, print $-1$.

If there is more than one solution, any one will do.

-----Explanation of Sample Inputs-----
In the first example, the substrings before rearrangement are “tral”, “rala”, “alal”, “lala”, and “alal”. Note that “alal” appears twice. After the rearrangement, all substrings are different.

In the third example, all substrings are different already in the input, so it suffices to print the original string.

-----Examples-----
Sample Input 1:
tralalal
Sample Output 1:
allatral

Sample Input 2:
zzzz
Sample Output 2:
-1
user: Brandon Greg Jr. considers an array to be scary if its leftmost element is the median of its elements. Given an array with distinct elements $[p_1, p_2, \ldots , p_ n]$, Brandon wants to count the number of scary subarrays.

A subarray is some contiguous chunk of elements $[p_ l, p_{l+1}, \ldots , p_ r]$ where $l\le r$. The median of a set of $n$ numbers is the middle number in sorted order if $n$ is odd, or the average of the middle two numbers in sorted order if $n$ is even. Note that all subarrays of length $1$ are scary, and no even-length subarrays are scary because the elements are all distinct.

-----Input-----
The first line of input contains a single integer $n$ ($1\le n\le 2\cdot 10^5$), representing the length of the given array.

The second line of input contains $n$ space-separated integers $p_ i$ ($1\le p_ i\le n$), representing the given array. It is guaranteed that the $p_ i$’s are distinct.

-----Output-----
Output one integer representing the number of scary subarrays.

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

Sample Input 2:
6
3 2 1 6 4 5
Sample Output 2:
8
user: Consider packing widgets for shipping where widgets cannot be stacked upon each other (2D packing). Each widget has a $1\times ~ l$ footprint and is $1$ unit high.

Boxes are available in any $W$ by $H$ by $1$ size such that $H/2 \leq W \leq 2H$, with $W$ and $H$ being integers. The company wants to minimize the amount of packing material that will be needed to fill empty squares in a box.

Given $N$, the number of widgets to be shipped, what is the smallest number of squares that will be left empty when those widgets are packed for shipping?

-----Input-----
Input consists of one line containing an integer $N$, the number of widgets to be packed, where $1 \leq N \leq 10^{16}$.

-----Output-----
Print a single line containing an integer denoting the minimum number of empty squares.

-----Examples-----
Sample Input 1:
47
Sample Output 1:
1

Sample Input 2:
523
Sample Output 2:
2
user: You are developing a new app intended to simplify expense-sharing among groups of friends. This app will allow them to keep track of who encountered an expense and how it should be shared with others through the form of IOUs. For instance, if Alice pays for a meal shared with Bob and Carol, and Bob’s and Carol’s shares were $\$ 5$ and $\$ 10$, respectively, then Bob would issue an IOU for $\$ 5$ to Alice and Carol would issue an IOU for $\$ 10$ to Alice.

Your app will maintain a ledger of who owes whom. Note that cycles can occur: For instance, if Bob initially owes Alice $\$ 10$ and later pays a $\$ 5$ expense on behalf of Alice, Alice would issue an IOU for $\$ 5$ to Bob. This IOU would then cancel out, or reduce, the IOU Alice holds from Bob from $\$ 10$ to $\$ 5$. It’s also possible for cycles to involve more than $2$ people.

Your app will be given a list of IOUs issued and settle them as much as possible by considering all cycles and reducing each debt in a cycle by the minimum amount of debt occurring in the cycle. After all cycles are considered and canceled, your app should output who owes whom how much. If there are multiple ways in which cancelation can occur, you may choose any of them as long as there are no cycles left at the end. However, you may not introduce IOUs between friends that never gave an IOU to each other, e.g., if Alice owes Bob money, and Bob owes the same amount to Carol, you cannot remove Bob from the picture and declare that Alice now owes Carol.

-----Input-----
The input consists of a single test case. The first line contains two integers $n$ and $m$ ($1 \le n \le 100, 0 \le m \le 10000$), where $n$ denotes the number of friends and $m$ denotes the number of IOUs issued. Friends are numbered $0$ to $n-1$. This is followed by $m$ lines containing three integers $a$, $b$, $c$ ($0 \le a < n, 0 \le b < n, a \ne b, 0 < c \le 1000$) denoting an IOU given by friend $a$ to friend $b$ for $c$ dollars. Any friend $i$ holds at most one IOU from any friend $j$ ($i \ne j$), but friend $i$ may hold an IOU from friend $j$ at the same time that friend $j$ holds an IOU from $i$.

-----Output-----
First, output a single number $p$, denoting the number of IOUs left after canceling all cycles. Then, on the following $p$ lines, output the IOUs that are left in the same form in which they appear in the input (e.g. using $3$ integers $a$, $b$, $c$ denoting that friend $a$ owes friend $b$ $c$ dollars). Do not include any IOUs fully canceled, i.e., all the IOUs you output must have $c > 0$.

-----Examples-----
Sample Input 1:
4 5
0 1 10
1 2 10
0 3 10
3 2 10
2 0 20
Sample Output 1:
0

Sample Input 2:
2 2
0 1 20
1 0 5
Sample Output 2:
1
0 1 15