meetforever/GRPO_CL huggingface dataset:
user: Near the great desert of Byteland lies a technologically advanced kingdom of hamsters. In the year 2020, to protect better their territories, the hamsters have decided to build a great wall surrounding crucial parts of their kingdom.

The kingdom of hamsters is represented by a convex polygon with $n$ vertices. Vertices are numbered from $1$ to $n$, either in clockwise or counter-clockwise order. The $i$-th vertex’s coordinate is $(x_ i, y_ i)$. The hamsters want to choose $6$ unique vertices (among the $n$ vertices) and build a convex hexagonal wall connecting these $6$ vertices. Furthermore, the hamsters want their wall to be as long as possible. In other words, the hexagonal wall should have largest circumference.

The hamsters want to know, for each vertex of their kingdom, if they choose this vertex as one of the vertices of the hexagonal wall (along with $5$ other vertices), what is the longest wall they can build?

Even though the hamsters are technologically advanced, their computers crashed when they executed a brute force code for this problem. Please help them!

-----Input-----
 - The first line of the input contains a single integer $n$ $(6 \le n \le 2000)$.
 - In the next $n$ lines, the $i$-th line contains two integers $x_ i$ and $y_ i$$(-10^9 \leq x_ i, y_ i \leq 10^9)$.

-----Output-----
Print exactly $n$ lines, the $i$-th line contains the maximum circumference of the convex hexagonal wall, such that the hexagon has the $i$-th vertex.

Your answer will be considered correct if its relative or absolute error doesn’t exceed $10^{-3}$.

Namely: let’s assume that your answer is $a$, and the answer of the jury is $b$. The checker program will consider your answer correct, if $\frac{|a-b|}{max(1,b)} \leq 10^{-3}$.

-----Explanation of the sample input-----
The left figure shows the first sample. There is only one way to create a convex hexagon. Its circumference is $2 + 4 \cdot \sqrt{2}$.

The right figure shows the second sample. The hexagon with the maximum circumference containing the first vertex is shown below. Its circumference is $6 + 2 \cdot \sqrt{29} + 4 \cdot \sqrt{2}$. Due to the symmetry of the polygon, the hexagons with the maximum circumference containing other vertices should have the same result.

-----Examples-----
Sample Input:
6
1 2
1 3
2 4
3 3
3 2
2 1
Sample Output:
7.656854249492381
7.656854249492381
7.656854249492381
7.656854249492381
7.656854249492381
7.656854249492381
user: Note that this is an easier version of the problem fancy

Oskar really loves onions. Unfortunately, the onions he loves belong to the local vegetable farmer, Lökas Tomatcek. Since Oskar is very cheap, he simply steals Lökas’ onions from the large, convex, onion field. For obvious reasons, Lökas does not approve of this. Rather than going to the police, however, Lökas wants to raise a high-energy laser fence around the onions to replace his wooden fence.

As laser fences are rather expensive, Lökas can only afford to upgrade $K$ of the fence posts to laser fence posts. If a set of posts are upgraded, the lasers will protect all the onions lying strictly within the convex hull of the selected fence posts. This may not be all of the onions though. Lökas wonders which posts he should upgrade if he wants to protect as many of his $N$ onions as possible.

The original fence has the shape of a convex polygon with $M$ fence posts as vertices, and all the onions lie strictly within the fence.

-----Input-----
The first line of input consists of three space-separated integers $3 \le N \le 1000$, $3 \le M \le 100$ and $3 \le K \le 100$. These are the number of onions, the number of old fence posts, and the number of fence posts Lökas can upgrade.

Then follow $N$ lines with two space-separated integers $0 \le X_ i, Y_ i \le 10^9$ each. $(X_ i, Y_ i)$ gives the coordinate of the $i$-th onion.

Then follow $M$ lines with two space-separated integers $0 \le A_ i, B_ i \le 10^9$ each. $(A_ i, B_ i)$ gives the coordinate of the $i$-th fence post.

The fence posts will be given in clockwise order of the old fence. No three fence posts will lie on a line.

-----Output-----
Output should be a single integer; the maximum number of onions Lökas can protect.

-----Examples-----
Sample Input:
3 5 3
1 1
2 2
1 3
0 0
0 3
1 4
3 3
3 0
Sample Output:
2
user: You have some bags of coins. Each bag contains exactly $k$ coins. Exactly one bag contains only counterfeit coins (we’ll call this the fake bag), while all other bags contain only real coins. All real coins weigh exactly the same number of grams. All counterfeit coins weigh exactly the same number of grams. You don’t know the exact weights of a real or counterfeit coin. You do know a counterfeit coin is strictly heavier than a real coin, but you do not know exactly how much heavier it is. The weights of the coins are positive real numbers.

You have a scale which you can use at most $m$ times. The scale has a left and right side. To use the scale, you can place any number of coins, taken from any of the bags, on each side of the scale, as long as the total number of coins on the left and right sides are exactly equal. The scale will return a single real number $s$. If $s$ is zero, both sides of the scale weigh exactly the same. If $s$ is negative, the left side is $|s|$ grams heavier than the right side. If $s$ is positive, the right side is $s$ grams heavier than the left side. Coins can be reused multiple times for different weighings, and you are able to keep track of which bag each coin came from. You must specify beforehand all weighings you want to perform (so you cannot adjust what gets weighed in future trials based on the results of previous trials). After using the scale $m$ times, you would like to be able to determine which bag is the fake bag.

You are now wondering: given $m$ and $k$, what is the maximum number of bags for which you can always determine the fake bag? This number can get large, so output it modulo the large prime $998244353$.

-----Input-----
The single line of input contains two space-separated integers $m$ and $k$ ($1 \leq m, k \leq 10^6$), where $m$ is the number of weighings available to you and $k$ is the number of coins in each bag.

-----Output-----
Output a single integer, which is the maximum number of bags for which you can determine the fake bag in $m$ weighings, modulo the large prime $998244353$.




-----Sample Explanation-----
One way we can use $2$ weighings to determine the fake bag among $9$ bags, each containing $1$ coin, is as follows:
 - On the first weighing, put the coins from bags $1,2,3$ on the left, and the coins from bags $7,8,9$ on the right.
 - On the second weighing, put the coins from bags $1,4,7$ on the left, and the coins from bags $3,6,9$ on the right.

We can determine the fake bag as follows:
 - The first weighing tells us which group of bags $(1,2,3)$, $(4,5,6)$, $(7,8,9)$ contains the fake bag (e.g. if the left side is heavier, then group $(1,2,3)$ contains the fake bag, if both sides are equal, then group $(4,5,6)$ contains the fake bag, otherwise group $(7,8,9)$ contains the fake bag).
 - The second weighing will tell us which group of bags $(1,4,7)$, $(2,5,8)$, $(3,6,9)$ contains the fake bag. The resulting fake bag can be uniquely determined as a result.

-----Examples-----
Sample Input:
2 1
Sample Output:
9
user: Given $S$, a set of integers, find the largest $d$ such that $a + b + c = d$ where $a, b, c$ and $d$ are distinct elements of $S$.

-----Input-----
The input starts with an integer $1 \le N \le 4000$, the number of elements in $S$. It is followed by $N$ lines containing the elements of $s$, one per line. Each element of $S$ is a distinct integer between $-536870912$ and $+536870911$, inclusive.

-----Output-----
Output a single line containing the maximum $d$ as described in the statement. If no such $d$ exists, output a single line containing no solution.

-----Examples-----
Sample Input:
5
2
3
5
7
12
Sample Output:
12
user: Taboo is a popular party game. In this game one player, the Clue Giver, prompts his/her teammates to guess a keyword by giving clues. The Clue Giver is also given a list of taboo strings that must not appear in the clues. For example, if the keyword is “Bruce Lee”, the famous kung-fu star, then the taboo strings may be “actor”, “kung-fu”, “fighting”, “martial arts” and “The Game of Death” (Bruce Lee’s final film). The Clue Giver may try such clues as “Fist of Fury star” and “Jeet Kune Do master” to avoid the taboo. Taboo strings bring challenges and fun to the guessing game.

Short clues are preferred, but now you are interested in the opposite: what is the longest clue? Given $N$ taboo strings $s_1, \dots , s_ N$, what is the longest clue string $s$ such that none of $s_1, \dots , s_ N$ appears as a substring of $s$? For simplicity, all taboo strings and your clue are represented as binary strings consisting only of 0’s and 1’s.

-----Input-----
The first line contains an integer, $N$, the number of taboo strings ($1 \leq N \leq 15000$). The following $N$ lines each contains a non-empty binary string $s_ i$, for $1 \leq i \leq N$. The sum of lengths of $s_1, \dots , s_ N$ will be at most $200000$.

-----Output-----
If your clue can be arbitrarily long, output -1. Otherwise, output a line containing the longest binary string that does not contain $s_1, \dots , s_ N$ as a substring. If there is more than one such longest string, output the one that is also smallest in lexicographic order.

-----Examples-----
Sample Input:
5
00
01
10
110
111
Sample Output:
11
user: You are an experienced investor who is investing in multiple companies. At the end of the year you are expected to file your taxes, which requires you to track and report the total number of shares you held during any of your trading days during the year.

The records you have kept are sorted by company: for each company, you have a list of notes stating how many shares of that company you owned on a given day.

Your tax accountant has asked you to produce a list of the total number of shares across all companies you have held on each of the days on which you have traded and recorded your holdings.

-----Input-----
The first line of input contains an integer $C$ ($1 \le C \le 20$), the total number of companies of which you hold shares. Next will be $C$ blocks of records, one for each company.

Each record block starts with an integer $K$ ($1 \le K \le 50$), which is the number of records you have kept for this company. This is followed by $K$ lines of input, each containing two integers $N$ ($1 \le N \le 1000$) and $D$ ($1 \le D \le 365$), denoting that you held $N$ shares of this company on day $D$. The days in the input may be specified in any order.

-----Output-----
Print, on a single line, for each day for which you have at least one record, the total number of shares held on that day. The share totals shall be printed in increasing order of trading days.

-----Examples-----
Sample Input:
3
2
20 100
100 10
1
150 50
1
150 100
Sample Output:
100 250 320
user: Nikolaj works at a company that sells a large number of products. Each product has a product ID, which is a large integer. For error detection, in particular in connection with manual database entry, it would be useful if each product also had a single “check” digit between $1$ and $9$ that can be quickly computed from the product ID. Nikolaj has been tasked with implementing this. He recently solved the problem “Digit Product” on Open Kattis and considers using the procedure described there.

He recalls that the idea is to start with a positive integer $x$ and repeatedly multiply all its nonzero digits, until only a single digit is left. For instance, if $x$ is $808$ then the resulting digit is $8$, because $8 \cdot 8 = 64$, $6 \cdot 4 = 24$, and $2 \cdot 4 = 8$.

However, Nikolaj is unsure about using this method, because the distribution of resulting digits seems uneven to him. To determine how the distribution looks, he writes a program that given two integers $L$ and $R$ determines how many numbers in the interval $[L, R]$ result in each digit.

-----Input-----
A single line consisting of two integers $L$ and $R$ with $1 \leq L \leq R \leq 10^{15}$.

-----Output-----
Print $9$ integers $a_1, a_2, \ldots , a_9$ on a single line. The $i$th number $a_ i$ is the number of integers $x$ satisfying $L \leq x \leq R$ such that repeated multiplication of the nonzero digits of $x$ results in the digit $i$.

-----Examples-----
Sample Input:
50 100
Sample Output:
3 7 4 6 5 7 2 15 2
user: You are given two convex polygons $A$ and $B$. It is guaranteed that $B$ is strictly contained inside of $A$.

You would like to make a sequence of cuts to cut out $B$ from $A$. To do this, you draw a straight line completely through $A$ that is incident to one of the edges of $B$, which separates $A$ into two pieces. You cut along this line and discard the piece that doesn’t contain $B$. You repeat this until the piece that you have left is exactly B.

The cost of making a cut is equal to the length of the cut (i.e. the length of the line through the remainder of $A$). Given $A$ and $B$, find the minimum cost needed to cut $B$ out.

-----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 containing a single integer $a$ ($3 \le a \le 200$), which is the number of points in polygon $A$. Each of the next $a$ lines will contain two integers $x$ and $y$ ($-10^6 \le x,y \le 10^6$), which are the vertices of polygon $A$, in clockwise order. It is guaranteed that polygon $A$ will be convex.

The next line will contain a single integer $b$ ($3 \le b \le 200$), which is the number of points in polygon $B$. Each of the next $b$ lines will contain two integers $x$ and $y$ ($-10^6 < x,y < 10^6$), which are the vertices of polygon $B$, in clockwise order. It is guaranteed that polygon $B$ will be convex. It is also guaranteed that polygon $B$ will reside entirely within the interior of polygon $A$.

No three points, within a polygon or across polygons, will be collinear.

-----Output-----
Output a single floating point number, which is the minimum cost to cut $B$ out of $A$. To be considered correct, this number must be within a relative error of $10^{-6}$ of the judges’ answer.

-----Examples-----
Sample Input:
4
0 0
0 14
15 14
15 0
4
8 3
4 6
7 10
11 7
Sample Output:
40.0000000000
user: You are looking for a particular family photo with you and your favorite relatives Alice and Bob. Each family photo contains a line-up of $n$ people. On the photo you’re looking for, you remember that Alice, who is taller than you, was somewhere on your left from the perspective of the photographer. Also, Bob who is taller than both you and Alice, was standing somewhere on your right.

Since you have a large number of family photos, you want to use your computer to assist in finding the photo. Many of the photos are quite blurry, so facial recognition has proven ineffective. Luckily, the Batch Apex Photo Classifier, which detects each person in a photo and outputs the sequence of their (distinct) heights in pixels, has produced excellent results. Given this sequence of heights for $k$ photos, determine which of these photos could potentially be the photo you’re looking for.

-----Input-----
 - The first line contains $1\leq k\leq 1000$, the number of photos you have to process.
 - Then follow two lines for each photo.
 - The first line contains a single integer $3 \leq n \leq 3\cdot 10^5$, the number of people on this photo.
 - The second line contains $n$ distinct integers $1\leq h_1, \dots , h_ n \leq 10^9$, the heights of the people in the photo, from left to right.

It is guaranteed that the total number of people in all photos is at most $3\cdot 10^5$.

-----Output-----
 - On the first line, output the number of photos $k$ that need further investigation.
 - Then print $k$ lines each containing a single integer $1\leq a_ i \leq n$, the sorted indices of the photos you need to look at.

-----Examples-----
Sample Input:
1
3
2 1 3
Sample Output:
1
1
user: Peter is co-owner of the incredibly successful Pete and Pat’s Pitas and Pizzas and his sales are on fire! But unfortunately, so is his building, due to carelessly laid delivery boxes placed too close to Pete’s famous wood burning pizza oven. After sifting though the remnants, one of the few things Pete is able to salvage is a ledger book, but the only thing he can make out on the charred pages is the profit he made during the last month. The insurance company would like to know how many pitas and how many pizzas Pete actually delivered over that period. Pete does recall how much profit he makes on each of these products, so he’s pretty confident that he can determine how many of each were sold during the last month given the total profit. Well perhaps “confident” is not the exact word Peter is looking for – it’s more like clueless. Can you help Pete out? I’m sure there are some discount coupons in it for you, or at least a really cheap price on a used pizza oven.

-----Input-----
Input consists of a single line containing $3$ values $p_ t$ $p_1$ $p_2$, where $0 \leq p_ t \leq 10000.00$ is the profit for the month and $0 < p_1, p_2 \leq 100.00$ are the profits Pete makes on a pita ($p_1$) and on a pizza ($p_2$). All values are in dollars and cents.

-----Output-----
Display two integers: the number of pitas sold and the number of pizzas sold so that the total profit equals the value given. If there is more than one combination of pitas and pizzas that give the specified profit, list them all, one combination per line, listing the combination with the smallest number of pitas first, then the combination with the second smallest number of pitas, and so on. If there are no combinations of pizza and pita sales that realize the profit, output none.

-----Examples-----
Sample Input:
725.85 1.71 2.38
Sample Output:
199 162
user: Mirko has a chessboard with $N$ rows and just three columns. Slavica has written an integer on each field. Mirko has $K$ dominoes at his disposal, their dimensions being $2 \times 1$, and has to arrange all of them on the board without overlapping, in a way that each domino covers exactly two fields of the board. He can rotate the dominoes as he please

Help Mirko cover the largest sum of numbers possible with the dominoes!

-----Input-----
The first line of input contains the integer $N$ ($1 \le N \le 1000$), the number of rows, and $K$ ($1 \le K \le 1000$), the number of dominoes available.

Each of the following $N$ lines contains three integers written in the $i$’th row of the board. All numbers will be less than $10^6$ in absolute value.

-----Output-----
The first and only line of output must contain the maximal sum possible to cover with exactly $K$ dominoes.

-----Examples-----
Sample Input:
5 3
2 1 -1
1 3 2
0 2 3
2 1 1
3 3 0
Sample Output:
16
user: Odd and Even have had their share of fun times playing the good old prime game:

They start with an arbitrary natural number, and take turns either adding $1$ or dividing by a prime (assuming the result is still a natural number), and the one to reach $1$ is the winner. However, now that they have a new friend, Ingmariay, they have decided to expand the rules of the game to allow for three-player action:

Instead of determining a winner for each round of play, they instead score points; the lowest number each of them has claimed during the round is the amount of points they get. (If any of them did not have the opportunity to claim any numbers, the starting number will be their score for that round.) At the end of the day, the player with the fewest points wins. And to avoid bad blood between themselves, they have all agreed that each of them only will focus on minimizing their own scores, and that whenever a player can choose different numbers that will result in the same score, that player will choose the lowest of those numbers. They have also agreed on a fixed order of play: Odd $\rightarrow $ Even $\rightarrow $ Ingmariay $\rightarrow ...$, but they alternate who gets to start.

You recently missed one of their exciting evenings of play, because you had to make problems for the NCPC event. Fortunately for you, they had recorded the numbers and starting players for each round, and told you that since they always play optimally (minimizing their own score), you could use this to simulate the event for yourself. Oh joy!

As an example round, assume that Even is chosen as the starting player, and with the starting number $15$. Then Even claims $16$, Ingmariay $8$, Odd $4$, Even $2$ and Ingmariay $1$. Odd gets $4$ points, Even $2$ and Ingmariay $1$.

-----Input-----
The first line of input contains a single integer $n$ ($1 \leq n \leq 1000$), the number of rounds they played that evening.

Then follow $n$ lines each beginning with the first character of the name of the starting player (either ’O’, ’E’ or ’I’), followed by a space and then the starting number for that round, in the range $[1, 10000]$

Note: If the starting number is $1$, all players receive $1$ point for that round.

-----Output-----
Output a single line with the score at the end of the day for each of the three contestants, in the order “Odd”, “Even”, “Ingmariay”.

-----Examples-----
Sample Input:
1
O 4
Sample Output:
2 1 4
user: As captain of your space ship you have never encountered a more fierce enemy than the one you have snuck upon now. You immediately bring out the big phaser cannon hoping to take out the flagship before they discover you. There is no room for mistakes and the shot will have to be perfect if you are to stand any chance at all against the flagship of the enemy. 

You start charging the phaser beam and retrieve the room layout of the flagship from the archives. You are situated directly above the enemy, from where the layout of the flagship can be modeled by a two-dimensional map of the rooms of the flagship. In this map, each room is a rectangle with sides parallel to the $x$ and $y$ axes (rectilinear), and no two rooms intersect (not even in a single point).

The phaser beam is configured by giving a point $(x, y)$ and an angle $\vartheta $. The phaser beam will start at $(x, y)$ and travel a distance $\ell $ in the direction specified by $\vartheta $, causing severe damage to every room touched by the phaser beam. Due to this, you aim at hitting as many rooms as possible.

The phaser beam is almost fully charged and the only missing piece is an optimal configuration of the weapon. Unfortunately, it turns out to be harder than you expected. However, there are still ten seconds before the charging is completed and hence you decide to make a computer program to solve the problem.

-----Input-----
The first line of input consists of two integers $r$ and $\ell $ ($1 \le r \le 15$, $1 \le \ell \le 1000$) where $r$ is the number of rooms in the flagship and $\ell $ is the length of a shot of the phaser.

Then follow $r$ lines, each of which contains four integers $x_1$, $y_1$, $x_2$, $y_2$ ($0 \le x_1 < x_2 \le 1000$, $0 \le y_1 < y_2 \le 1000$), indicating that there is a room in the flagship with lower left corner $(x_1, y_1)$ and upper right corner $(x_2, y_2)$.

-----Output-----
Output one line with the maximum number of rooms that can be hit by one phaser beam. Recall that if the beam touches a room it is counted as a hit.

You may assume that the answer is numerically stable in the following sense: if all rooms are expanded by a distance of $10^{-6}$ in all four directions, the answer does not change.

-----Examples-----
Sample Input:
5 8
2 1 4 5
5 1 12 4
5 5 9 10
1 6 4 10
2 11 7 14
Sample Output:
4