meetforever/GRPO_CL huggingface dataset:
user: Near a military base there is a system of trenches, modeled as line segments on a plane. During nighttime, when most soldiers are fast asleep, three guards stand watch of the trenches. Two guards can see each other if there is a trench (or a row of trenches) along the entire straight line segment between them and there is no third guard on that line segment.

For security reasons, the guards must be placed so that each guard sees the other two. How many ways can they be placed?

-----Input-----
The first line contains the integer $N$ $(1 \leq N \leq 20)$, the number of trenches. Each of the next $N$ lines contains the description of one trench: four non-negative integers $X_1, Y_1, X_2, Y_2$ (all less than or equal to $1000$), where $X_1$ and $Y_1$ are coordinates of one end, while $X_2$ and $Y_2$ are coordinates of the other end of the trench.

Trenches in the input may overlap and share endpoints.

-----Output-----
Output the number of ways the guards can be placed on a single line.

-----Examples-----
Sample Input 1:
6
0 0 1 0
0 0 0 1
1 0 1 1
0 1 1 1
0 0 1 1
1 0 0 1
Sample Output 1:
8

Sample Input 2:
4
5 1 7 1
1 1 5 1
4 0 4 4
7 0 3 4
Sample Output 2:
1
user: You are an architect and you have just been appointed to build a new swimming hall. The organisation behind these plans has acquired funding for a swimming pool and surrounding building as large as they want, but unfortunately they could not find anyone willing to pay for the floor surrounding the pool. They decided to pay for the floor tiles out of their own pocket. Because this has already cost them an arm and a leg, they want you to use all the floor tiles in your proposed plan.

Being an architect, you care for aesthetics. You see it as absolutely vital that both the swimming pool and the surrounding building are perfect squares. This raises an interesting problem: how can you make sure that the square shapes are guaranteed, while still using all the floor tiles the organisation bought?

Given the number of tiles $n$, find the length of the side of the building $m$ and and the length of the side of the pool $k$ such that $n=m^2-k^2$, or print impossible if no suitable $m$ and $k$ exist.

-----Input-----
 - One line containing a single integer $1\leq n\leq 10^9$.

-----Output-----
Print two non-negative integers $m$, $k$ such that $n = m^2 - k^2$, or print impossible if no such integers exist. If there are multiple valid solutions, you may output any one of them.

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

Sample Input 2:
10
Sample Output 2:
impossible
user: Mirko got an array of integers for his birthday from his grandmother Norma. As any other kid, he was hoping for some money, but got an array. Luckily, in his town there is a pawn shop that buys up arrays. The cost of an array of integers is $\min \cdot \max \cdot L$ kunas, where $\min $ is the minimal integer in the array, $\max $ is the maximal and $L$ is the array length. Mirko is going to sell a subsequence of consecutive numbers from his array. He calculated the average price of all such subsequences.

In order to check his result, he wants you to do the same. He will be pleased with only the last $9$ digits of the sum of all prices, so you don’t need to bother with large and real numbers.

-----Input-----
The first line of input contains an integer $N$ ($1 \leq N \leq 500000$).

Each of the following $N$ lines contains a member of Mirko’s array. The members of the array will be integers from the interval $[1, 10^8 ]$.

-----Output-----
The first and only line of output must contain an integer, the last $9$ digits of the required sum from the task. You don’t need to output the leading zeroes of that 9-digit integer.

-----Examples-----
Sample Input 1:
2
1
3
Sample Output 1:
16

Sample Input 2:
4
2
4
1
4
Sample Output 2:
109
user: After their pool burst, Mirko and Slavko started collecting cards. In their neighbourhood, card collection is taken seriously and there are strict rules for the purchase and trading of cards.

Purchasing cards is always done by two children together. Each of them gives half the required funds and two cards are bought. Then they race to the fountain downtown, the winner getting both cards. If they arrive at the exact same time, each of them gets one of the cards.

At first the rules performed okay, but soon accusations started that some kids could not have acquired their cards only through such purchases.

One day all the children met to figure out if there were irregularities. They managed to agree on the exact number of cards each of them currently has. They also made a partial list of who went to the store with whom, but they do not know who won any of the races and took the cards on those occasions.

Write a program that determines the participants in all purchases that were made and who won the subsequent races so that after all the purchases, the cards counts correspond to the given counts. Assume that before any purchases, the children had no cards.

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

-----Input-----
The first line contains the integers $N$ and $M$ ($1 \le N \le 100$, $0 \le M \le 1000$), the number of children and the number of purchases they recall. The children are labeled $1$ through $N$.

The second line contains $N$ integers, the number of cards each child currently has. The total number of cards of all children is at most $2000$.

The following $M$ lines contain two integers $a \ne b$ each, the labels of the children who made the purchase.

-----Output-----
On the first line, output the total number of purchases.

Each of the following lines should describe one purchase. The description of a purchase consists of three numbers: the two distinct labels of the children that made the purchase and the number $0$, $1$ or $2$, how many cards the first child got after the race.

Note: A solution will always exist, although it will not necessarily be unique.

-----Explanation of Sample Input 1-----
In the first example, there are only two children, labeled $1$ and $2$. The first child should end up with five cards, the second with one.

After the first purchase, each of the children got one card.

After the second and third purchases, the first child got both cards both times.

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

Sample Input 2:
4 3
5 3 1 1
1 3
2 3
4 1
Sample Output 2:
5
1 3 1
2 3 2
4 1 0
2 4 1
3 1 0
user: The robot revolution is finally here, albeit not quite in the highly explosive way envisioned in various science fiction books and movies. It seems that, perhaps due to a small typo in the AI source code, the robots are not taking our lives but instead taking our livelihoods. One early job market fatality in this revolution was the (somewhat niche) occupation as jeopardy player: already in 2011 the Watson computer defeated two legendary but inferior human jeopardy champions. 

Nowadays, more and more of Jeopardy’s viewers are AIs themselves and as such the show is considering having categories on topics that are more popular with this new number-crunching viewer base. Focus group testing has revealed that AIs are particularly fond of the “Binomial Coefficients” category. The premise of this category is that the answer that contestants get is some positive integer $X$, and the contestants must respond with a question of the form “What is $n$ choose $k$?” (and this is a correct response if the binomial coefficient $n$ choose $k$ equals $X$).

Write an AI to play this new Jeopardy category. If there are several different possible solutions for $n$ and $k$, the AI should choose the most elegant solution, having the smallest value of $n$, and of those with the smallest $n$ it should choose the one with the smallest value of $k$.

-----Input-----
Input consists of a single integer $X$ ($1 \le X \le 10^{100}$).

-----Output-----
Output two non-negative integers $n$ and $k$ such that the binomial coefficient $n$ choose $k$ equals $X$, with ties between multiple solutions broken as explained above.

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

Sample Input 2:
2020
Sample Output 2:
2020 1
user: Little Mirko is studying the hash function which associates numerical values to words. The function is defined recursively in the following way:
 - $f( \text {empty word} ) = 0$
 - $f( \text {word} + \text {letter} ) = ( ( f( \text {word} ) \cdot 33 ) \oplus \mathrm{ord}( \text {letter} ) ) \mathbin {\% } \mathrm{MOD}$

The function is defined for words that consist of only lowercase letters of the English alphabet. $\oplus $ stands for the bitwise exclusive or operator (i.e. $0110 \oplus 1010 = 1100$), $\mathrm{ord}(\text {letter})$ stands for the ordinal number of the letter in the alphabet ($\mathrm{ord}(\mathtt{a}) = 1$, $\mathrm{ord}(\mathtt{z}) = 26$) and $A \mathbin {\% } B$ stands for the remainder of the number $A$ when performing integer division with the number $B$. $\mathrm{MOD}$ will be an integer of the form $2^M$.

Some values of the hash function when $M$ = 10:
 - $f( \mathtt{a} ) = 1$
 - $f ( \mathtt{aa} ) = 32$
 - $f ( \mathtt{kit} ) = 438$

Mirko wants to find out how many words of the length $N$ there are with the hash value $K$. Write a programme to help him calculate this number.

-----Input-----
The first line of input contains three integers $N$, $K$ and $M$ ($1 \leq N \leq 10$, $0 \leq K < 2^M$, $6 \leq M \leq 25$).

-----Output-----
The first and only line of output must consist of the required number from the task.

-----Examples-----
Sample Input 1:
1 0 10
Sample Output 1:
0

Sample Input 2:
1 2 10
Sample Output 2:
1
user: Web pages are littered with ads (advertisements) these days. You wrote a simple program to detect images which are ads and remove them.

Characters that are any of the following are allowed in images:
 - alphanumeric characters
 - question mark ‘?’
 - exclamation mark ‘!’
 - comma ‘,’
 - full stop ‘.’
 - whitespace ‘ ’

Any images contained characters that are not allowed are considered ads, and will be removed by your program. The ‘$+$’ character will be used only as the border of an image.

All images will follow these rules:
 - They are all axis-aligned rectangles with ‘$+$’ as its border and is at least $3 x 3$ big (including the borders).
 - All ‘$+$’ characters belong to the border of exactly 1 image.
 - Borders of different images will not be adjacent to each other (contact at their corners is also not allowed).
 - They can contain other images while adhering to the rules above.
 - For every banned character that is found inside at least one image, in order not to remove too many unnecessary parts of the images on the web page, you decide that the smallest image (in terms of pixel count) that contains that character will be flagged as an ad and removed.
 - When an ad is removed, every character inside the ad and the border is replaced with a whitespace.

Note that the web page is not considered an image. Characters found on the web page that is not in any image should not be removed.

-----Input-----
The first line of the input contains $1 \leq H \leq 200$ and $1 \leq W \leq 200$, the height and width of the web page to be processed respectively. $H$ lines follow, each line contain exactly $W$ ASCII characters whose ASCII decimal value fall in the range $[32, 126]$.

-----Output-----
Output the web page with all the ads removed. Do not print extra spaces or new lines.

-----Subtasks-----
 - (21 Points): There is no ads in the web page at all.
 - (21 Points): There is either one image or no image in the web page.
 - (28 Points): No images will be found inside other images.
 - (30 Points): No additional constraint.

-----Examples-----
Sample Input 1:
8 20
 apples are great!  
++++++++++++++++++++
+ oranges are      +
+ way better!      +
+ #oranges>apples  +
++++++++++++++++++++
 check out our      
 fresh apples!      
Sample Output 1:
 apples are great!  
                    
                    
                    
                    
                    
 check out our      
 fresh apples!      

Sample Input 2:
7 7
+++++++
+  =  +
+ +++ +
+ + + +
+ +++ +
+     +
+++++++
Sample Output 2:
user: After passing the final round of the Hunter Exam, Gon is now a Hunter! But before continuing his exciting journey, Gon decides to visit his home in ‘While Island’.

Aunt Mito is happy that Gon has become a Hunter and has visited home. But at the same time, she feels worried about Gon — the life of a Hunter is challenging. Thus, she will not allow Gon to leave ‘While Island’, unless Gon is able to solve a puzzle prepared by aunt Mito.

But what puzzle should Mito give to Gon? Mito opened ‘IMO 2019 Problems’ and found the following problem:

The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Prof. Tuy has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k > 0$ coins showing $H$, then he turns over the $k$-th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n = 3$ the process starting with the configuration $THT$ would be: $THT \rightarrow HHT \rightarrow HTT \rightarrow TTT$, which stops after three operations.

Show that, for each initial configuration, Prof. Tuy stops after a finite number of operations.

‘This problem is too easy’ — Mito thought. ‘Gon will be able to solve it in no time’.

After some more thinking, Mito changed the above problem to the following harder version:

For each initial configuration $C$, let $L(C)$ be the number of operations before Prof. Tuy stops.

Given a sequence $S$ consisting of $H$, $T$ and $?$, representing some initial state, where $?$ denotes an unknown state. Calculate the average value of $L(C)$ over all possible sequences $C$ represented by $S$.

Even though Gon is smart, he is still unable to solve this question. Please help Gon!

-----Input-----
The input contains a single non-empty line with at most $10^6$ characters — the sequence $S$.

-----Output-----
The output must contain a single number — the average value of $L(C)$, as explained in the statement.

Your answer will be considered correct if its relative or absolute error doesn’t exceed $10^{-6}$.

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 $\frac{|a-b|}{max(1,b)} \leq 10^{-6}$.

-----Examples-----
Sample Input 1:
HH
Sample Output 1:
2.0

Sample Input 2:
H?
Sample Output 2:
1.5
user: Josip is a strange painter. He wants to paint a picture consisting of $N \times N$ pixels, where $N$ is a power of two ($1, 2, 4, 8, 16$ etc.). Each pixel will be either black or white. Josip already has an idea of how each pixel will be coloured.

This would be no problem if Josip’s painting process wasn’t strange. He uses the following recursive process:
 - If the picture is a single pixel, he colours it the way he intended.
 - Otherwise, split the square into four smaller squares and then:
 - Select one of the four squares and colour it white.
 - Select one of the three remaining squares and colour it black.
 - Consider the two remaining squares as new paintings and use the same three-step process on them.

Soon he noticed that it was not possible to convert all his visions to paintings with this process. Your task is to write a program that will paint a picture that differs as little as possible from the desired picture. The difference between two pictures is the number of pairs of pixels in corresponding positions that differ in colour.

-----Input-----
The first line contains an integer $N$ ($1 \leq N \leq 512$), the size of the picture Josip would like to paint. $N$ will be a power of $2$. Each of the following $N$ lines contains $N$ digits $0$ or $1$, white and black squares in the target picture.

-----Output-----
On the first line, output the smallest possible difference that can be achieved. On the next $N$ lines, output a picture that can be painted with Josip’s process and achieves the smallest difference. The picture should be in the same format as in the input.

Note: The second part of the output (the picture) may not be unique. Any correct output will be accepted.

-----Examples-----
Sample Input 1:
4
0001
0001
0011
1110
Sample Output 1:
1
0001
0001
0011
1111

Sample Input 2:
4
1111
1111
1111
1111
Sample Output 2:
6
0011
0011
0111
1101
user: An LCD panel is composed of a grid of pixels, spaced $1$ alu (“arbitrary length unit”) apart both horizontally and vertically. Wires run along each row and each column, intersecting at the pixels. Wires are numbered beginning with $1$ and proceeding up to a panel-dependent maximum. The vertical wire numbered $1$ lies along the left edge of the panel, and the horizontal wire numbered $1$ lies along the bottom edge of the panel.

A pixel will activate, turning dark, when a current is present along both the vertical and horizontal wire passing through that pixel.

For a period of time, we will send pulses of current down selected wires. The current flows down the wires at a speed of one alu per atu (“arbitrary time unit”). The pulses themselves have a length measured in atus. A pixel activates when current is passing through both intersecting wires at the same time. If the leading edge of a pulse on one wire reaches the intersection at the exact same time that the trailing edge of a pulse on the other wire leaves that intersection, the pixel is not activated.

All pulses in vertical wires start from the bottom of the grid. All pulses in horizontal wires start from the left of the grid. At most one pulse will travel along any one wire.

Given the schedule of pulses to be sent through the wires, determine how many pixels will have been activated by the time all pulses have exited the top and right of the grid.

-----Input-----
The first line contains $n$, the number of current pulses, with $1\le n\le 200000$.

Following this are $n$ lines, each describing a single pulse. Each such line contains four elements, separated from one another by a single space:
 - A single character that is either ‘h’ or ‘v’, indicating the horizontal/vertical direction of the pulse.
 - An integer $t$, $1\le t\le 200000$, denoting the starting time of the pulse. The starting time is considered to be the moment when the leading edge of a vertical [horizontal] pulse crosses horizontal [vertical] wire #$1$.
 - An integer $m$, $1\le m\le 200000$, denoting the length of the pulse.
 - An integer $a$, $1\le a\le 100000$, denoting the wire number (horizontal or vertical) along which the pulse travels.

-----Output-----
Print on a single line the number of pixels that will have activated by the time the last pulse of current has left the grid.

-----Examples-----
Sample Input 1:
4
h 1 4 1
v 2 4 2
h 10 2 2
v 11 2 3
Sample Output 1:
2

Sample Input 2:
4
h 1 10 1
h 5 10 2
v 1 10 1
v 5 10 3
Sample Output 2:
4
user: Librarian Jurica has $N$ shelves in his library, and each shelf can contain $M$ books. Jurica is a good librarian so he decided to make an inventory in the library and, if it’s necessary, return the books that aren’t in their place to their right place. He moves the books in the following way:
 - moving the books one place to the left or to the right on a shelf if the place to the left or to the right is available,
 - taking a book from a shelf and placing it to an available place on that or any other shelf.

Careful Jurica can’t move books if he has a book in his hands. Additionally, he can’t take more than one book at once.

Jurica has been having back pains ever since he had to move all the volumes of the printed edition of Wikipedia from the first to the second floor so now he wants to put all the books in place with as little lifting as possible because his back is hurting. What is the minimal number of lifting he needs?

-----Input-----
The first line of input contains the integers $N$ and $M$ ($1 \leq N \leq 1000$, $1 \leq M \leq 1000$).

Each of the following $N$ lines contains $M$ integers, the $i$-th line describing the current state of the $i$-th shelf.

Number $0$ denotes an empty place on the shelf, and a number different than $0$ denotes that there is a book in that place denoted with that number. All books are denoted with different numbers from $1$ to $K$, where $K$ is the total number of books on the shelves. After that, an additional $N$ lines follow, each containing $M$ integers, the $i$-th line describing the wanted state of the $i$-th shelf.

In the initial and final state of the shelves, the same books will appear.

-----Output-----
The first and only line of output must contain the required minimal number of lifting or $-1$ if it is impossible to arrange the books in the aforementioned way.

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

Sample Input 2:
3 3
1 2 3
4 5 6
7 8 0
4 2 3
6 5 1
0 7 8
Sample Output 2:
4
user: Zapray recently did not have much time to finish his homework due to his addiction to Pokemon. What is even worse, this week’s algorithm class has assigned a very challenging homework problem: “Given an array of integers, find the longest consecutive sub-array so that each element in the sub-array appears exactly twice.”

Zapray remembers how to find the longest consecutive sub-array so that each element in the sub-array appears exactly once using sliding window. But he could not figure out how to extend it for this problem. Could you help Zapray?

-----Input-----
The first line of the input has an integer $N$ ($1 \leq N \leq 10^5$), the length of the array. The second line contains $N$ positive integers giving the elements of the array. Each integer is no larger than $10^5$.

-----Output-----
Output the length of the longest sub-array so that each element in the sub-array appears exactly twice.

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

Sample Input 2:
8
1 2 1 3 1 3 1 2
Sample Output 2:
4
user: Mirko is a huge fan of chess and programming, but typical chess soon became boring for him, so he started having fun with rook pieces.

He found a chessboard with $N$ rows and $N$ columns and placed $K$ rooks on it. Mirko’s game is made of the following rules:
 - Each rook’s power is determined by an integer.
 - A rook sees all the fields that are in his row or column except its own field.
 - We say that a field is attacked if the binary XOR of all the powers of the rooks that see the field is greater than $0$.

Notice that the field a rook is at can be attacked or not.

Initially, Mirko placed the rooks in a certain layout on the board and will make $P$ moves. After each move, determine how many fields are attacked. Every rook can be moved to any free field on the whole board (not only across column and row).

-----Input-----
The first line of input contains integers $N$, $K$, $P$ ($1 \leq N \leq 1\ 000\ 000\ 000$, $1 \leq K \leq 100\ 000$, $1 \leq P \leq 100\ 000$). Each of the following $K$ lines contains three integers $R$, $C$, $X$ ($1 \leq R, C \leq N$, $1 \leq X \leq 1\ 000\ 000\ 000$) which denote that initially there is a rook of power $X$ on the field $(R, C)$. Each of the following $P$ lines contains four integers $R_1$, $C_1$, $R_2$, $C_2$ ($1 \leq R_1, C_1, R_2, C_2 \leq N$) which denote that the rook has moved from field $(R_1, C_1 )$ to field $(R_2, C_2 )$. It is guaranteed that there will not be two rooks on one field at any point.

-----Output-----
The output must consist of $P$ lines, the $k$-th line containing the total number of attacked fields after $k$ moves.

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

Sample Input 2:
2 2 2
1 1 1
2 2 2
2 2 2 1
1 1 1 2
Sample Output 2:
4
2