meetforever/GRPO_CL huggingface dataset:
user: The Swedes wanted to arrange a ping pong tournament among the $n$ high school students in the nation. Since Sweden is the fairest country on earth, they wanted to ensure that the ranking of all the players is perfectly fair. Therfore, they decided that every high school student in the nation should play against every other high school student. Sadly, when all the matches were done, it turned out that ranking the players in a fair manner was hard, or even impossible; sometimes it was the case that a player $a$ had beat some other player $b$, player $b$ had beat some player $c$ and player $c$ had beat player $a$. Regardless of which ranking the Swedish experts suggested, someone was treated unfairly in the sense that they were ranked below someone they had beaten. The Swedish king thus decided to disqualify some players, so that there would be no ambiguity in the ranking among the remaining players. His experts then came up with a set $S$ of $k$ players to disqualify. Unfortunately, you are one of these disqualified players.

Together with the other disqualified players, you plead with the king to make him undo his decision. While he remains firm in the belief that his approach will yield a fair ranking, he has offered your group a deal: If you can find a set $S’$ containing strictly less than $k$ players to disqualify instead of your own group, then the king will disqualify this group instead. The group $S$ of disqualified players has now bestowed upon you the responsibility of finding that different group $S’$. They require that none of the originally disqualified players are in $S’$, and that the group you suggest is as small as possible.

-----Input-----
The first line contains two positive integers, $n$ the number of high school students, and $k$ the number of disqualified players. Then follows $n$ lines, each containing $n$ integers. The $j$-th integer on the $i$-th line is $1$ if player $i$ beat player $j$, and $0$ otherwise (i.e. there is also a $0$ at the $i$-th entry of the $i$-th line, even though player $i$ didn’t lose to himself). Finally, there is a line with $k$ integers indicating the set $S$, the players who were disqualified. Each player is identified by a number between $0$ and $n-1$. You may assume that, when the disqualified players $S$ are removed, there exists a fair ranking of the remaining players. A fair ranking is one such that nobody is ranked below someone they beat. We remark that even though the Swedish experts couldn’t find a fair ordering of all the players, there might still be one. Note: The input may be quite large.

We always have $2 \leq k \leq n \leq 2000$.

-----Output-----
Output a single integer, the size $k’$ of the smallest set $S’$ of players needed to be disqualified to make a fair ranking. Recall that none of the originally disqualified players of $S$ can be in $S’$, and $k’$ must be strictly less than $k$. If no such solution exists, print “impossible”.

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

Sample Input 2:
4 2
0 0 1 1
1 0 0 1
0 1 0 0
0 0 1 0
1 2
Sample Output 2:
impossible
user: Mirko is trying to debug a piece of his code. First he creates an array of $N$ integers and fills it with zeros. Then he repeatedly calls the following C++ procedure:

void something( int jump ) {
  int i = 0;
  while( i < N ) { seq[i] = seq[i] + 1; i = i + jump;
  }
}

As you can see, this procedure increases by one all elements in the array whose indices are divisible by jump.

Mirko calls the procedure exactly $K$ times, using the sequence $X_1,X_2,X_3, \ldots , X_ k$ as arguments.

After this, Mirko has a list of $Q$ special parts of the array he needs to check to verify that his code is working as it should be. Each of these parts is defined by two numbers, $L$ and $R$ ($L \leq R$) the left and right bound of the special part. To check the code, Mirko must compute the sum of all elements of seq between and including $L$ and $R$. In other words $\mathtt{seq}[L] + \mathtt{seq}[L+1] + \mathtt{seq}[L+2] + \ldots + \mathtt{seq}[R]$. Since he needs to know the answer in advance in order to check it, he asked you to help him.

-----Input-----
The first line of input contains two integers, $N$ ($1 \leq N \leq 10^6$), the size of the array, and $K$ ($1 \leq K \leq 10^6$), the number of calls to something that Mirko makes.

The second line contains $K$ integers: $X_1,X_2,X_3, \ldots ,X_ k$, the arguments passed to the procedure ($1 \leq X_ i < N$).

The third line contains one integer $Q$ ($1 \leq Q \leq 10^6$), the number of special parts of the array Mirko needs to check.

The next $Q$ lines contain two integers each $L_ i$ and $R_ i$ ($0 \leq L_ i \leq R_ i < N$), the bounds of each special part.

-----Output-----
The output should contain exactly $Q$ lines. The $i$-th line should contain the sum of elements $\mathtt{seq}[L_ i] + \mathtt{seq}[L_ i+1] + \mathtt{seq}[L_ i+2] + \ldots + \mathtt{seq}[R_ i]$.

-----Examples-----
Sample Input 1:
10 4
1 1 2 1
3
0 9
2 6
7 7
Sample Output 1:
35
18
3

Sample Input 2:
11 3
3 7 10
3
0 10
2 6
7 7
Sample Output 2:
8
2
1
user: A binary search tree is a tree in which every node has at most two children nodes (a left and a right child). Each node has an integer written inside it. If the number $X$ is written inside a node, then the numbers in its left subtree are less than $X$ and the numbers in its right subtree are greater than X. You will be given a sequence of integers between 1 and $N$ (inclusive) such that each number appears in the sequence exactly once. You are to create a binary search tree from the sequence, putting the first number in the root node and inserting every other number in order.

When inserting a new number $Y$ in a tree, you first traverse the tree as if you were searching for $Y$. When you arrive at a node $Z$ such that you can’t continue searching for $Y$, you put $Y$ as a left or right son of $Z$ depending on if $Z>Y$ or $Z<Y$, so that after the insertion the tree will still be a binary search tree. After the insertion you add the depth of $Y$ to a counter $C$ and print the value of $C$. The counter $C$ is set to $0$ in the beginning.

-----Input-----
The first line contains the integer $N$ $(1 \leq N \leq 300000)$, the length of the sequence.

The remaining $N$ lines contain the numbers in the sequence, integers in the interval $[1, N]$. The numbers will be distinct.

-----Output-----
Output $N$ integers each on its own line, the values of the counter $C$ after each number is inserted into the tree.

-----Examples-----
Sample Input 1:
4
1
2
3
4
Sample Output 1:
0
1
3
6

Sample Input 2:
5
3
2
4
1
5
Sample Output 2:
0
1
2
4
6
user: You have most definitely heard the legend of King Arthur and the Knights of the Round Table. Almost all versions of this story proudly point out that the roundness of the Round Table is closely related to Arthur’s belief of equality among the Knights. That is a lie! In fact, Arthur’s choice of table is conditioned by his childhood traumas.

In fact, Arthur was forced to clean up quadratic tables from a young age after a tournament in pick-up sticks1 had been played on them. After the tournament, typically there would be a bunch of sticks on the table that do not touch each other. In the spirit of the game, the organizers issued strict regulations for the table cleaners. More precisely, the sticks on the table need to be removed one by one in a way that the cleaners pull them in the shortest way towards the edge of the table closest to where they are currently sitting. They also mustn’t rotate or touch the other sticks while doing this (not even in the edge points).

In this task, we will represent the table in the coordinate system with a square that has opposite points in the coordinates $(0,0)$ and $(10000, 10000)$, whereas the sticks will be represented with straight line segments that lie within that square. We will assume that Arthur is sitting at the edge of the table lying on the $x$-axis. Then the movement of the stick comes down to translating the line segment along the shortest path towards the $x$-axis until the stick falls off the table (as shown in the image). It is your task to help Arthur determine the order of stick movements that meets the requirements from the previous paragraph.

-----Input-----
The first line of input contains the integer $N$ ($1 \leq N \leq 5000$), the number of sticks on the table. Each of the following $N$ lines contains four integers $x_1$, $y_1$, $x_2$, $y_2$ ($0 \leq x_1, y_1, x_2, y_2 \leq 10000$) that denote the edge points of a stick.

-----Output-----
The first and only line of output must contain space-separated stick labels in the order which they need to be taken off the table. A stick’s label corresponds to its position in the input sequence.

If there are multiple possible solutions, output any of them.

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

Sample Input 2:
4
0 0 1 1
1 2 0 3
2 2 3 3
4 0 3 1
Sample Output 2:
4 3 1 2
user: John, a student who is taking the game development course, recently developed a mobile game called Fruit Slicer for his coursework. In the game the player slices fruits that are throw into the air by swiping the touch screen. However the game is quite simple because John was not able to write code for the geometry required for a more complex version. In the game each slice is a straight line of infinite length, and all fruits have the same shape of a circle with unit radius. The figure shows a cool snapshot of John’s game. 

John introduces his game to his best friend Sean, who soon gets bored of playing the simple game. But as a teaching assistant of the algorithm course, Sean decides to turn the game into a homework assignment. He asks the students in the algorithms course to write a program that can compute the best slice at any given moment of the game. Given the locations of the fruits, the program should determine the maximum number of fruits that can be sliced with a single straight-line swipe.

As a student in Sean’s class, you are now the one who is facing this challenge.

-----Input-----
The first line has a single integer $n$ ($1 \leq n \leq 100$). The next $n$ lines each have two real numbers giving the $x$ and $y$ coordinates of a fruit. All coordinates have an absolute value no larger than $10^4$ and are given with exactly two digits after the decimal point. Fruits may overlap.

-----Output-----
Output the maximum number of fruits that can be sliced with one straight-line swipe. A swipe slices a fruit if the line intersects the inner part or the boundary of the fruit.

-----Examples-----
Sample Input 1:
5
1.00 5.00
3.00 3.00
4.00 2.00
6.00 4.50
7.00 1.00
Sample Output 1:
4

Sample Input 2:
3
-1.50 -1.00
1.50 -1.00
0.00 1.00
Sample Output 2:
3
user: A distant country has $N$ cities in it. The elections were just held in this country, so the new prime minister was elected. Currently, there is not a single road in this country, so the prime minister decided to modernize the country by connecting some cities with two-way highways and form counties. Two cities will be located in the same county if it is possible to get to one city from the other using the newly built roads. Each city will be located in exactly one county. Each county consists of one or more cities.

The cities are represented as points in a two-dimensional coordinate system. The road between two cities is represented as a line segment connecting two points where the cities are located. The length of the road is equal to the length of the line segment in kilometers.

The country is currently suffering from recession, so the prime minister decided that, because of the lack of budget, they will not build roads longer than $D$ kilometers. Additionally, the prime minister is happy about the small things, so he will be happy if, in at least one county, there exists a nonempty subset of cities (it can include all cities in the county) where the total sum of residents is divisible by $K$. For instance, if $K = 4$ and there is a county with cities that have $3$, $5$, $7$ residents respectively, the prime minister will be happy because the sum of residents in the first two cities is equal to $8$.

Help the prime minister in cutting the costs by determining the minimal $D$ such that the prime minister can build roads and be happy about the small things at the same time.

-----Input-----
The first line of input contains the integers $N$ and $K$ ($1 \leq N \leq 50000$, $1 \leq K \leq 30$). Each of the following $N$ lines contains three integers $x_ i$, $y_ i$, $k_ i$ ($0 \leq x_ i, y_ i, k_ i \leq 100000000$), that represent the x coordinate of the city, the y coordinate and the number of residents in that city, respectively. There will not be two cities with the same coordinates in the input data. Additionally, there will not be a single city with the number of residents divisible by $K$.

-----Output-----
The first and only line of output must contain the minimal $D$ such that it is possible to build roads with the condition that the prime minister is happy. An answer accurate to $3$ decimal places will be accepted. The input data will be such that there is always a solution.

-----Examples-----
Sample Input 1:
3 3
0 4 4
1 5 1
2 6 1
Sample Output 1:
1.414

Sample Input 2:
6 11
0 0 1
0 1 2
1 0 3
1 1 4
5 5 1
20 20 10
Sample Output 2:
5.657
user: David and Martin retired from their jobs and became shepherds. After each winter they need to rebuild the sheep pen. Each year they have a different collection of straight fence segments of various lengths. Naturally, they want to build a sheep pen that covers the largest area. They always spend a lot of time choosing, rearranging, and placing the segments but they are never sure whether they got the optimal solution. Please help them.

-----Input-----
The only input line starts with $n$, the number of fence segments and then contains a list of $n$ integers, the lengths (in meters) of the fence segments. The lengths are integers between $1$ and $100$. The number of fence segments $n$ is between $3$ and $80$.

-----Output-----
The output contains one line with the maximum area (in square meters) of a polygon whose sides have lengths given by the input. Not all segment lengths listed need to be used. Answers with an absolute error of at most $0.005$ will be accepted. If no sheep pen can be built, output $0$.

-----Examples-----
Sample Input 1:
4 1 1 1 1
Sample Output 1:
1.0

Sample Input 2:
3 1 1 1
Sample Output 2:
0.433
user: Barica is an unusual frog. She lives in a pond where $N$ plants float on the surface of the water. The plants are numbered $1$ through $N$. When viewing from above, the location of each plant is given by a pair of coordinates. What makes Barica unusual is her fear of jumping diagonally and in the negative direction. More precisely, she can jump from a plant at coordinates $(x_1, y_1)$ to another at coordinates $(x_2, y_2)$ only if:
 - $x_2 > x_1$ and $y_2 = y_1$, or
 - $y_2 > y_1$ and $x_2 = x_1$

For each plant, we know the number of flies in its immediate vicinity. Barica can use her swift tongue to eat all flies near the plant she is at.

Barica absorbs one energy unit for each fly she eats, and uses $K$ energy units for each jump she makes. Barica can not make a jump if she doesn’t have enough energy units beforehand.

Barica wants to go from plant $1$ to plant $N$ and have the largest amount of energy possible after arriving. Barica initially has no energy and must gather energy for her first jump from the flies around plant 1.

Find the sequence of plants Barica should travel to achieve her goal.

-----Input-----
The first line of input contains two integers $N$ and $K$ ($2 \le N \le 300000$, $1 \le K \le 1000$) separated by a space.

Each of the following $N$ lines contains three integers $X$, $Y$ and $F$ ($0 \le X, Y \le 100000$, $0 \le F \le 1000$) separated by spaces, meaning that there is a plant at coordinates $(X, Y)$ with $F$ flies around it.

The first plant in the input is plant $1$, the second is plant $2$ etc.

No two plants will share the same pair of coordinates.

Note: The input data will guarantee that a sequence of jumps, although not necessarily unique, will always exist.

-----Output-----
Output the final energy level on the first line. Output an integer $L$, the number of plants Barica should travel, including plants $1$ and $N$. On the following $L$ lines, output the sequence of plants Barica should travel.

-----Examples-----
Sample Input 1:
6 5
1 1 5
2 1 5
1 2 4
2 3 5
3 2 30
3 3 5
Sample Output 1:
5
4
1 1
2 1
2 3
3 3

Sample Input 2:
8 10
1 1 15
2 2 30
1 2 8
2 1 7
3 2 8
2 3 7
4 2 100
3 3 15
Sample Output 2:
36
5
1 1
1 2
2 2
3 2
3 3
user: “Arrange” is a planetary popular Flash game. In “Arrange” the player is given a permutation of numbers $1$ to $N$ and a list of allowed swaps. He then has to perform a sequence of swaps that transforms the initial permutation back to the ordered sequence $1,2,3,4,5, \ldots , N$.

In order to break the high score list, you need to perform the minimum amount of swaps possible. You can’t do that, but you can write a program that does it for you!

-----Input-----
The first line of input contains two integers, $N$ ($2 \le N \le 11$), the length of the initial sequence and $M$ ($1 \le M \le N(N – 1) / 2$), the number of allowed swaps.

The second line of input contains a permutation of the numbers $1$ to $N$.

The next $M$ lines contain descriptions of allowed swaps. Each such line contains two distinct numbers $1 \le A < B \le N$, indicating that you are allowed to swap the $A$-th number in the sequence with the $B$-th number. The input never contains two identical swaps.

You may assume that the input is such that a solution exists.

-----Output-----
Output the minimum possible number of swaps to transform the permutation to $1, 2, \ldots , N$.

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

Sample Input 2:
3 2
2 1 3
1 3
2 3
Sample Output 2:
3
user: Peter’s favorite season is winter. Caught up in the festivities, Peter would like to decorate his city.

The city has many areas for things like ice skating, skiing, and snowball fights. There are roads connecting some pairs of areas. Peter would like to put a special decoration along each of these roads. There are three types of decorations of costs $0$, $1$, and $2$, respectively.

Peter would like his decoration to meet some properties:
 - For any pair of distinct roads adjacent to an area, let $a$ and $b$ be the costs of decorations along those roads. It must hold that $(a+b) \bmod 3 \neq 1$.
 - The sum of costs along any cycle must be an odd number.

A cycle is a sequence of areas connected by roads that form a loop. Each area may appear exactly once in the loop.

Peter would like to know: What is the cheapest amount he can pay to decorate his city the way he wants?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will begin with a line with two integers $n$ and $m$ ($1 \le n,m \le 10^5$), where $n$ is the number of areas and $m$ is the number of roads. The areas are numbered $1..n$.

Each of the next $m$ lines will each contain two integers $a$ and $b$ ($1 \le a < b \le n$), indicating that there is a road directly connecting areas $a$ and $b$. No two roads will connect the same two areas. It may or may not be possible to get from every area to every other area along the roads.

-----Output-----
Output a single integer, which is the minimum cost of decorating the city, or $-1$ if it isn’t possible to decorate the city according to Peter’s properties.

-----Examples-----
Sample Input 1:
5 8
1 4
4 5
1 5
1 2
1 3
2 3
3 5
2 5
Sample Output 1:
-1

Sample Input 2:
6 5
2 4
3 5
1 5
3 6
1 6
Sample Output 2:
5
user: Canyons are deep ravines between escarpments or cliffs. They exist on more than just Earth. For example, Valles Marineris on Mars is a vast canyon system running along the Martian equator and is roughly the size of the United States.

Working for a prestigious mapping company, you’ve been tasked with making maps for such canyons. A canyon can be represented in 2D by a simple polygon outline. The maps you will be constructing will be perfectly square and axis aligned. This is due to the mapping company’s desire that tops of their maps are always North. In order to increase the detail, sometimes multiple maps are used to cover just one canyon. Such a collection is called a mapping system. The mapping system of the canyon must cover the entire area of the canyon. The maps in the mapping system must also be the same scale with respect to the canyon and the same size with respect to their printing. This allows the maps to be easily compared when viewed together.

Your mapping system will have exactly $k$ maps. You need to find a mapping system that completely covers the canyon, but each map covers as little area as possible, since a map with less area can be shown at a higher detail. All of the maps must be perfectly square, and must cover the same amount of area on the canyon. The maps can overlap. Since area is the square of side length, just output the side length for simplicity. If things go well enough, your mapping system may even be used to map Valles Marineris in the near future.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each input begins with a line with two space-separated integers $n$ ($3 \le n \le 2000$) and $k$ ($1 \le k \le 3$), where $n$ is the number of vertices in the polygon, and $k$ is the number of square maps in the mapping system. The next $n$ lines each contain two space-separated integers $x$ $y$ ($-20000 \le x, y \le 20000$). These are the coordinates of the polygon, in order. No two edges of the polygon will intersect. All points will be distinct. No three consecutive points will be collinear. The polygon will be a simple polygon which does not touch or cross itself. It will not be degenerate, and will have positive area.

-----Output-----
Output a real number rounded to exactly two decimal places, representing the minimum side length with respect to the canyon for each square map in your mapping system, where the maps are identically sized, as small as possible, and the system still covers the entire canyon.

-----Examples-----
Sample Input 1:
4 1
1 1
5 1
5 5
4 2
Sample Output 1:
4.00

Sample Input 2:
6 3
-8 -8
0 -1
8 -8
1 0
0 10
-1 0
Sample Output 2:
9.00