meetforever/GRPO_CL huggingface dataset:
user: You are competing in a ski slalom, and you need to select the best skis for the race. The format of the race is that there are $N$ pairs of left and right gates, where each right gate is $W$ metres to the right of its corresponding left gate, and you may neither pass to the left of the left gate nor to the right of the right gate. The $i$-th pair of gates occurs at distance $y_{i}$ down the hill, with the horizontal position of the $i$<sup>th</sup> left gate given by $x_{i}$. Each gate is further down the hill than the previous gate (i.e. $y_{i}$ < $y_{i+1}$ for all $i$). 

You may select from $S$ pairs of skis, where the $j$-th pair has speed $s_{j}$. Your movement is governed by the following rule: if you select a pair of skis with speed $s_{j}$, you move with a constant downward velocity of $s_{j}$ metres per second. Additionally, at any time you may move at a horizontal speed of at most $v_{h}$ metres per second.

You may start and finish at any two horizontal positions. Determine which pair of skis will allow you to get through the race course, passing through all the gates, in the shortest amount of time.

-----Input Specification-----
The first line of input contains the three integers $W$, $v_{h}$, and $N$, separated by spaces, with $1 <= W <= 10^8$, $1 \leq v_{h} \leq 10^6$, and $1 \leq N \leq 10^5$.

The following $N$ lines of input each contain two integers $x_{i}$ and $y_{i}$, the horizontal and vertical positions respectively of the $i$-th left gate, with $1 \leq x_{i},y_{i} \leq 10^8$.

The next line of input contains an integer $S$, the number of skis, with $1 \leq S \leq 10^6$.

The following $S$ lines of input each contain one integer $s_{j}$, the speed of the $j$-th pair of skis, with $1 \leq s_{j} \leq 10^6$.

-----Output Specification-----
If it is impossible to complete the race with any pair of skis, print the line IMPOSSIBLE. Otherwise, print the vertical speed $s_{j}$ of the pair of skis that allows you to get through the race course in the shortest time.

-----Examples-----
Sample Input:
3 2 3
1 1
5 2
1 3
3
3
2
1
Sample Output:
2
user: Ewan is a park ranger, and part of his duty is to observe how the ecosystem changes over time. He has just detected that a group of pine trees and aspens have started to share the same space. Before, there was a clear line between the species.

This isn’t unusual, but Ewan is a bit curious about how this will evolve in the future. To track it, he wants to know how much of the area is covered by both species right now.

-----Input-----
The input begins with a single line with two integers $P$ and $A$: the number of pine trees and aspens, respectively. Then follow $P + A$ lines, one line for each tree. The first $P$ of these lines contain locations for the pine trees, and the remaining $A$ lines contain locations for the aspens.

Each tree is represented by two real numbers $x_ i$ and $y_ i$, representing the location of the tree on a 2D map in metres.

-----Output-----
Output the area covered by both species in square metres. Your answer must have an absolute or relative error of at most $10^{-3}$.

A point is covered by a species if it is at the edge of or inside a triangle made by three distinct trees of that species.

-----Limits-----
 - $0 \leq P, A \leq 1000$
 - $0.0 \leq x_ i, y_ i \leq 1000.0$
 - $0.2 < |x_ i-x_ j|+|y_ i-y_ j|$ for all $0 \leq i < j < P + A$
 - Real numbers in the input will have at most $8$ digits after the decimal point

-----Examples-----
Sample Input:
3 3
0.0 6.0
6.0 0.0
6.0 6.0
4.0 4.0
10.0 4.0
4.0 10.0
Sample Output:
4.0
user: No wonder the old bookcase caved under the massive piles of books Tom had stacked on it. He had better build a new one, this time large enough to hold all of his books. Tom finds it practical to have the books close at hand when he works at his desk. Therefore, he is imagining a compact solution with the bookcase standing on the back of the desk. Obviously, this would put some restrictions on the size of the bookcase, it should preferably be as small as possible. In addition, Tom would like the bookcase to have exactly three shelves for aesthetical reasons. 

Wondering how small his bookcase could be, he models the problem as follows. He measures the height $h_ i$ and thickness $t_ i$ of each book $i$ and he seeks a partition of the books in three non-empty sets $S_1,S_2,S_3$ such that $\left(\sum _{j=1}^3 \max _{i \in S_ j} h_ i \right) \cdot \left( \max _{j=1}^3 \sum _{i\in S_ j} t_ i \right)$ is minimized, i.e., the area of the bookcase as seen when standing in front of it (the depth needed is obviously the largest width of all his books, regardless of the partition). Note that this formula does not give the exact area of the bookcase, since the actual shelves cause a small additional height, and the sides cause a small additional width. For simplicity, we will ignore this small discrepancy.

Thinking a moment on the problem, Tom realizes he will need a computer program to do the job.

-----Input-----
The input begins with a line containing a single positive integer $N$, $3\leq N\leq 70$ giving the number of books. Then $N$ lines follow each containing two positive integers $h_ i$, $t_ i$, satisfying $150\leq h_ i \leq 300$ and $5\leq t_ i \leq 30$, the height and thickness of book $i$ respectively, in millimeters.

-----Output-----
Output one line containing the minimum area (height times width) of a three-shelf bookcase capable of holding all the books, expressed in square millimeters.

-----Examples-----
Sample Input:
4
220 29
195 20
200 9
180 30
Sample Output:
18000
user: Filoména wants to write a message on her newly opened sticker store. She wants to write the message using stickers that she sells in the store. Each sticker spells a word and has a price. She has unlimited supply of any type of sticker. The stickers can overlap but the maximum thickness of the message is two stickers, that is, at no place should more than two stickers overlap (otherwise the message would stick out too much). Now she wonders whether it is possible to write the message and if so, what is the minimal cost of doing it?

-----Input-----
The first line contains the message written as one word in capital letters. The second line contains $n$, the number of different types of stickers. Each of the next $n$ lines contains the word spelled by the sticker and the price of the sticker (the word is a single word in capital letters; the price of the sticker is a positive integer $\leq 100000$). You may assume that $n\leq 500$ and that length of each sticker is at most $20$. The length of the message is at most $1000$.

-----Output-----
The output consists of a single line. If the message can be assembled from the stickers with no more than two stickers overlapping at any place then it should contain the minimal cost of creating the message. Otherwise it should contain the string IMPOSSIBLE.

-----Examples-----
Sample Input:
BUYSTICKERS
4
BUYER 10
STICKY 10
TICKERS 1
ERS 8
Sample Output:
28
user: “Next to the confrontation between two highly honed batteries of lawyers, jungle warfare is a stately minuet.” ­–Bill Veeck.

Frivolous lawsuit or legitimate litigation? That is the question the judge Kattis faces today as she presides over lawsuits between individuals and corporations.

There are $R$ individuals and $S$ corporations involved in lawsuits that Kattis must preside over today. For simplicity, we label the individuals $1, 2, \dots , R$ and the corporations $1, 2, \dots , S$. Each of the individuals and corporations is involved in at least one of the $L$ lawsuits; each lawsuit involves exactly one individual and one corporation, and in particular, the $i^\text {th}$ of these lawsuits involves the individual $A_ i$ and the corporation $B_ i$.

There has been a growing resentment among the populace that judges tend to be biased, and Kattis really wants to be known as an equitable and fair judge. For each lawsuit, she needs to rule in favor of one party or the other; the party she rules in favor of is said to have “won” the lawsuit. In an aggressive bid to improve her optics, she has decided to ignore the merits of the arguments involved (as the public is incapable of nuanced discussion anyway), but to rule in such a way that no single individual or corporation wins too many lawsuits, so as to avoid accusations of bias.

That is, she wants to make rulings with the goal of minimizing the maximum number of lawsuits any single individual or corporation wins.

What ruling should she make for each lawsuit?

-----Input-----
The first line of input contains three integers, $R$ ($1 \leq R \leq 2000$), $S$ ($1 \leq S \leq 2000$) and $L$ ($\max (R, S) \leq L \leq 2000$) the number of individuals, the number of corporations and the number of lawsuits.

The next $L$ lines contain the descriptions of the lawsuits. In particular, the $i^\text {th}$ of these lines contains two integers $A_ i$ ($1 \leq A_ i \leq R$) and $B_ i$ ($1 \leq B_ i \leq S$), denoting that the $i^\text {th}$ lawsuit involves individual $A_ i$ and corporation $B_ i$.

It is guaranteed that each individual and corporation is involved in at least one lawsuit. Note that it is possible for the same pair of one individual and one corporation to be involved in more than one lawsuit.

-----Output-----
Output $L$ lines, where the $i^\text {th}$ of these lines contains either INDV$A_ i$ or CORP $B_ i$, describing the party Kattis should rule in favor of for the $i^\text {th}$ lawsuit, such that the maximum number of lawsuits any single individual or corporation wins is minimized.

If there are multiple correct answers, you can output any of them.

-----Examples-----
Sample Input:
5 3 8
1 1
2 1
3 1
4 2
5 2
3 3
4 3
5 3
Sample Output:
INDV 1
INDV 2
CORP 1
INDV 4
CORP 2
INDV 3
CORP 3
INDV 5
user: “Phew, that was the last one!” exclaimed the garden helper Tim as he threw the last tree plant to the ground. His employer, countess Esmeralda Hunt who owned the estate, has ordered him to arrange an avenue of aspen trees along both sides of the front road leading up to the house. The first trees in the avenue are supposed to be planted at the very beginning of the road, and the last trees of the avenue at the very end of the road. Tim, who was constantly being reminded of her sense of accuracy, knew that the countess would insist on the trees being placed in perfectly aligned tree pairs, one on each side of the road, and with exactly the same spacing between the pairs along the road. However, when bringing the tree plants to the estate, Tim had just dropped them arbitrarily along the left side of the road and was now facing the task of moving the trees to their correct positions to meet the countess’s requirements. Being stronger in mind than in arms, and since it was time for a coffee break anyway before he started digging, he sat down to figure out which trees to move to which positions so as to minimize the total (Euclidean) distance that he had to move the trees. 

-----Input-----
The input file starts with a positive even integer $N$ between $4$ and $2000$ (inclusive), giving the total number of trees in the avenue. The next line contains two integers $L$ and $W$, where $1 \le L \le 10000$ is the length of the road, in meters, and $1 \le W \le 20$ is the width of the road, in meters. The next $N$ lines each describe where Tim had dropped of the trees. Each such line contains an integer $0 \le p \le L$ indicating the position of a tree plant along the left side of the road, measured in meters from the start of the road.

-----Output-----
Output the smallest total number of meters the tree plants need to be moved. The answer should be given with an absolute or relative error of at most $10^{-6}$.

-----Examples-----
Sample Input:
4
10 1
1
0
10
10
Sample Output:
2.4142135624
user: Bwahahahahaha!!! Your nemesis, the dashingly handsome spy Waco Powers, has at last fallen to your secret volcano base’s deathtraps (or so you assume, being a little too busy to witness it firsthand). At long last, you are all set to Conquer The World!

Nothing will stand in your way! Well, nothing except a minor problem of logistics. Your evil armies have announced that they will not continue carving their relentless path of destruction across the puny nations of the world without being paid. And unfortunately you are running low on cash – a volcano lair has many wonderful qualities, but “reasonably affordable” is not one of them. You have had to pull funds from the travel budget to pay your ungrateful underlings. Now you are not sure how you will actually get your armies into position to Conquer The World.

You have a map of the nations of the world and all your available transport routes between them. Each route connects two nations and has a fixed cost per army that uses it. The routes are laid out such that there is exactly one way to travel between any two nations. You know the current position of each of your armies and how many you will need to place permanently in each nation in order to subjugate it. How can you move the armies into place as cheaply as possible so you can Conquer The World?

-----Input-----
The first line of input contains an integer $n$ ($1 \le n \le 250000$), the number of nations. This is followed by $n-1$ lines, each containing three integers $u$, $v$, and $c$ ($1 \le u, v \le n$, $1 \le c \le 10^6$), indicating that there is a bidirectional route connecting nations $u$ and $v$, which costs $c$ per army to use.

Finally, another $n$ lines follow, the $i^{\text {th}}$ of which contains two non-negative integers $x_ i$ and $y_ i$, indicating that there are currently $x_ i$ armies in nation $i$, and you need at least $y_ i$ armies to end up in that nation in the final configuration. The total number of armies (the sum of the $x_ i$ values) is at least the sum of the $y_ i$ values, and no more than $10^6$.

-----Output-----
Display the minimum cost to move your armies such that there are at least $y_ i$ armies in nation $i$ for all $i$.

-----Examples-----
Sample Input:
3
1 2 5
3 1 5
2 1
5 0
1 3
Sample Output:
15
user: Pacman is cleaning his ice rink! The rink, like everything in PacMan’s universe, wraps! In other words, if you go off the edge of the rectangular rink to the right, you’ll end up in the same row on the left side of the rink. Go off of the left side of the rink, and you’ll end up on the right. Going up off the top of the rink causes you to end up in the same column but on the bottom. Go off of the bottom, and you’ll end up at the top.

What makes this zamboni special is that when it cleans the ice, it leaves a color on the ice! At the start, the ice is colorless (white). Whenever the zamboni is on a cell of ice, it overwrites the color on that cell. Each color is represented by a capital letter of the alphabet. The colors are in alphabetical order. The zamboni switches to the next letter in the alphabet at each step, and wraps at the end of the alphabet. If the current color is P, the next color is Q. If the current color is Z, the next color is A.

PacMan uses a very specific algorithm to clean the ice:

stepSize <- 1
loop numSteps times Move stepSize steps in the current direction Rotate the direction of the zamboni 90 degrees clockwise Switch to the next color stepSize <- stepSize + 1
end loop

The zamboni is originally facing up. Knowing where the zamboni starts, the size of the rink, and the number of steps taken, determine what the ice looks like after the zamboni has completed its run.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each input consists of a single line with $5$ space-separated integers, $r\ c\ i\ j\ n$, where:
 - $r$ ($1 \le r \le 2000$) is the number of rows of the ice rink
 - $c$ ($1 \le c \le 2000$) is the number of columns of the ice rink
 - $i$ ($1 \le i \le r$) is the starting row of the zamboni
 - $j$ ($1 \le j \le c$) is the starting column of the zamboni
 - $n$ ($1 \le n \le 10^{18}$) is the number of steps (numSteps in PacMan’s algorithm)

Note that row $1$ is the top row, and column $1$ is the left column.

-----Output-----
Output the state of the ice rink after the zamboni has completed its run, as a $r \times c$ grid of characters. Each character should be a capital letter (A-Z) representing the color of the ice, or a dot (.) if the ice is white, or an ’at’ sign (@) if the cell is the final location of the Zamboni.

-----Examples-----
Sample Input:
5 5 3 3 4
Sample Output:
.....
..BBC
..A.C
....C
@DDDD
user: Arranging Hat is a cushy job indeed; high impact work, absolute authority, and 364 days of holiday every year. However, the hat has decided that it can do even better—it would like very much to become a tenured professor. 

Recently the hat has been reading computer science papers in its ample spare time, and of course, being an arranging hat, it is particularly interested in learning more about sorting algorithms.

The hat’s new contribution is to a class of algorithms known as lossy sorting algorithms. These usually work by removing some of the input elements in order to make it easier to sort the input (e.g., the Dropsort algorithm), instead of sorting all the input.

The hat is going to go one better—it is going to invent a lossy sorting algorithm for numbers that does not remove any input numbers and even keeps them in their original place, but instead changes some of the digits in the numbers to make the list sorted.

The lossiness of the sorting operation depends on how many digits are changed. What is the smallest number of digits that need to be changed in one such list of numbers, to ensure that it is sorted?

-----Input-----
The input consists of:
 - one line containing the integers $n$ and $m$ ($1 \le n \le 40$, $1 \le m \le 400$), the number of numbers and the number of digits in each number, respectively.
 - $n$ lines each containing an integer $v$ ($0 \le v < 10^{m}$). The numbers are zero-padded to exactly $m$ digits.

-----Output-----
Write a sorted version of the array, after making a minimum number of digit changes to make the numbers sorted (the numbers must remain zero-padded to $m$ digits). If there are multiple optimal solutions, you may give any of them.

-----Examples-----
Sample Input:
5 3
111
001
000
111
000
Sample Output:
001
001
001
111
200
user: The gear ratio designers ate BikeCo need your help! They have designed a new set of gear ratios $n_1/d_1, n_2/d_2, \cdots , n_{12}/d_{12}$ for a bicycle with front and rear derailleurs having two and six sprockets respectively. Your task is to figure out, if possible, the sprocket sizes needed to achieve the required gear ratios. 

The sprockets of the front derailleur are connected to the pedals and drive the rear sprockets by chain. You recall from Sprocket Science $101$ that a cyclist uses derailleurs to select any combination of front and rear sprocket. The resulting gear ratio is $u/v$ where $u$ is the front sprocket sprocket size and $v$ is the rear sprocket size.

You are constrained to exactly two sprockets on the front derailleur and six on the rear derailleur, leading to at most $12$ possible gear ratios. Note that the resulting gear ratios are not necessarily unique.

-----Input-----
Twelve lines with two integers $n_ i, d_ i$ ($1 \le n_ i, d_ i \le 100$) separated by forward slash, where $n_ i/d_ i$ is the $i$th gear ratio.

-----Output-----
If the specified gear ratios can be achieved, output the required sprocket sizes on two lines with space-separated integers. The first line has the two sprocket sizes for the front derailleur, and the second line gives the six sprocket sizes needed for the rear derailleur. Due to engineering limitations no sprocket size may exceed $10000$. If there are multiple such sets of sprocket sizes that generate the specified gear ratios, output any one of them.

If no solution exists, print impossible.

-----Examples-----
Sample Input:
19/13
10/1
19/14
4/3
20/7
19/7
20/13
19/15
10/7
20/17
19/2
19/17
Sample Output:
19 20
17 15 14 13 7 2
user: Mia is a newly hired administrative assistant. Though Mia was introduced to everyone in the company on her first day, she is forgetful and struggles to remember people’s names. Being too shy to ask for names again, she discovers a way to recall people’s names during lunchtime without asking.

Mia orders lunch for all of her colleagues in each of the next several days. On any day, Mia orders burgers for some of her colleagues and salads for the rest of her colleagues. The number of burgers she orders may vary per day. After placing the order, she sends an email to the colleagues who get a burger for lunch, and also to the remaining colleagues about their salads. Mia has an email list with all her colleagues’ names. She may choose by name who gets a burger and who gets a salad. Mia can see her colleagues as they are eating their lunch. Thus, by observing who turns out to eat a burger in the office and who eats a salad, she may gain some information to help her uniquely identify the names of her colleagues.

For example, suppose there are three colleagues with names Alice, Danielle, and Jennifer, and Mia can order one burger plus two salads on each day. On the first day, if Mia orders a burger for Alice and salads for Danielle and Jennifer, she can then tell who is Alice by observing who eats a burger. On the second day, Mia may order a burger for Danielle and a salad for Jennifer (and a salad for Alice who is already identified). Consequently she can uniquely identify all three colleagues.

What is the maximum number of colleagues that Mia can uniquely identify in the next few days, if she allocates the burger and salad recipients optimally?

-----Input-----
The first line of input contains two space-separated integers $n$ ($2 \leq n \leq 30$) and $m$ ($1 \leq m \leq 10$), where Mia has $n$ colleagues and will be ordering lunch for $m$ days.

Each of the next $m$ lines contains a single integer $a$ ($1 \leq a < n$), which is the number of burgers Mia orders on that day. The days are listed in order.

-----Output-----
Output a single integer, which is the maximum number of Mia’s $n$ colleagues that she can uniquely identify after $m$ days.

-----Examples-----
Sample Input:
4 2
2
2
Sample Output:
4
user: Mirko and his faithful friend Slavko got really bored one day. The result of their boredom is the creation of a new game! In the beginning of the game, they draw $N$ points in a coordinate system. The players take turns and Mirko plays first. He draws a straight line which is parallel to one of the axes of the coordinate system and passes through one of the $N$ points. In the following moves, the player draws a straight line which is parallel to one of the axes of the coordinate system and passes through one of the $N$ points located on the line drawn in the previous move of the opponent. No single line must be drawn twice. The loser is the player who cannot play his move. Determine who has the winning strategy.

-----Input-----
The first and only line of input contains the positive integer $N$ ($1 \leq N \leq 10000$). Each of the following $N$ lines contains two integers $X$ and $Y$, the coordinates of the points drawn ($1 \leq X,Y \leq 500$).

-----Output-----
The first and only line of output must contain the name of the winner, either “Mirko” or “Slavko”.

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