meetforever/GRPO_CL huggingface dataset:
user: Frane has been given the task of sorting an array of numbers. The array consists of $N$ integers, each between $1$ and $N$ (inclusive), with each of those appearing exactly once in the array. Frane has come up with the following sorting algorithm which operates in $N$ phases, and named it turbosort:
 - In the first phase, the number $1$ is moved to position $1$ by repeatedly swapping consecutive elements.
 - In the second phase, the number $N$ is moved to position $N$ in the same manner.
 - In the third phase, the number $2$ is moved to position $2$.
 - In the fourth phase, the number $N-1$ is moved to position $N-1$.
 - And so on...

In other words, when the number of the phase is odd, Frane will choose the smallest number not yet chosen, and move it to its final position. In even phases he chooses the largest number not yet chosen. Write a program which, given the initial array, output the number of swaps in each phase of the algorithm.

-----Input-----
The first line contains an integer $N$ ($1 \le N \le 100000$), the number of elements in the array. Each of the following $N$ lines contains an integer between $1$ and $N$ (inclusive), the array to be sorted. The array will contain no duplicates.

-----Output-----
For each of the $N$ phases, output the number of swaps on a single line.

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

Sample Input 2:
5
5
4
3
2
1
Sample Output 2:
4
3
2
1
0
user: Three infinite lines define a triangle, unless they meet at a common point or some of them are parallel.

Given a collection of infinite lines, what is the largest possible perimeter of a triangle defined by some three lines in the collection?

-----Input-----
The first line of input contains a single integer $n$ ($3 \leq n \leq 100$) indicating the number of infinite lines.

The next $n$ lines describe the collection of infinite lines. The $i^{\textrm{th}}$ such line contains four integers $x_1, y_1, x_2, y_2$ ($-10000 \leq x_1, y_1, x_2, y_2 \leq 10000$) where $(x_1, y_1) \neq (x_2, y_2)$ are two points lying on the $i^{\textrm{th}}$ infinite line.

-----Output-----
Display a single real value which is the perimeter of the largest triangle that can be formed from three of the infinite lines. Your output will be considered correct if it is within an absolute or relative error of $10^{-5}$ of the correct answer.

If no triangle can be formed using the given lines, then you should instead display the message no triangle.

-----Examples-----
Sample Input 1:
3
0 0 0 1
0 0 1 0
0 1 1 0
Sample Output 1:
3.4142135624

Sample Input 2:
3
0 0 0 1
0 0 1 0
0 0 1 1
Sample Output 2:
no triangle
user: In order to build a staircase to a famous pagoda on top of a mountain, the local authority has identified $N$ positions along the mountain slope $a_1, a_2, \ldots , a_ N$ where $a_ i$ is the height of the $i$-th position and $a_ i \leq a_{i+1}$ for all $0<i<N$.

The cost to build a staircase from position $i$ to position $j$ is$\min _{v \in \mathbb {Z}}{\sum _{s=i}^{j}{|a_ s-v|^k}}$

To speed up the process of building the staircase from position $1$ to position $N$, the local authority has decided to give the job to $G$ builders to build the staircase in parallel. The sequence of $N$ positions will be divided into $G$ segments of consecutive positions where every position belongs to exactly one segment and each segment is managed by exactly one builder.

Given an integer $G$ ($1 \leq G \leq N$), your task is to identify a way to allocate jobs to $G$ builders to minimize their total building costs.

-----Input-----
 - The first line contains $3$ integers $N$, $G$, $k$ $(1 \leq N \leq 2000, \; 1 \leq G \leq N, \; 1 \leq k \leq 2)$.
 - The second line contains $N$ integers $a_1, a_2, \ldots , a_ N$$(1 \leq a_ i \leq 10^6, \; a_ i \leq a_{i+1} \; \forall \; 0<i<N)$.

-----Output-----
Print the minimum building cost required.

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

Sample Input 2:
5 1 2
1 2 3 4 5
Sample Output 2:
10
user: The grille cipher  is a technique that dates back to 1550 when it was first described by Girolamo Cardano. The version we’ll be dealing with comes from the late 1800’s and works as follows. The message to be encoded is written on an $n \times n$ grid row-wise, top to bottom, and is overlaid with a card with a set of holes punched out of it (this is the grille).

The message is encrypted by writing down the letters that appear in the holes, row by row, then rotating the grille 90 degrees clockwise, writing the new letters that appear, and repeating this process two more times. Of course the holes in the grille must be chosen so that every letter in the message will eventually appear in a hole (this is actually not that hard to arrange).

An example is shown below, where the message “Send more monkeys” is encrypted as “noeesrksdmnyemoj”, after adding a random letter to fill out the grid (this example corresponds to the first sample input.)

 Figure I.1 

If the message is larger than the $n \times n$ grid, then the first $n^2$ letters are written in the grid and encrypted, then the next $n^2$ are encrypted, and so on, always filling the last grid with random letters if needed. Here, we will only be dealing with messages of length $n^2$.

Your job, should you choose to accept it, is to take an encrypted message and the corresponding grille and decrypt it. And we’ll add one additional twist: the grille given might be invalid, i.e., the holes used do not allow every location in the grid to be used during the encryption process. If this is the case, then you must indicate that you can’t decrypt the message.

-----Input-----
The input starts with a line containing a positive integer $n\le 10$ indicating the size of the grid and grille. The next $n$ lines will specify the grille, using ‘.’ for a hole and ‘X’ for a non-hole. Following this will be a line containing the encrypted message, consisting solely of lowercase alphabetic characters. The number of characters in this line will always be $n^2$.



-----Output-----
Output the decrypted text as a single string with no spaces, or the phrase “invalid grille” if the grille is invalid.

-----Examples-----
Sample Input 1:
4
XX.X
X.X.
XXXX
.XXX
noeesrksdmnyemoj
Sample Output 1:
sendmoremonkeysj

Sample Input 2:
4
.XX.
XXXX
XXXX
.XX.
abcdefghijklmnop
Sample Output 2:
invalid grille
user: Robot Turtles is one of Theta’s favorite games. In this game, kindergarteners learn how to “code” by creating programs that move a turtle from a starting field to a diamond. Robot Turtles is reportedly the most successful game funded by the Kickstarter incubator.

In the actual board game, an adult plays the role of the “turtle master,” which is the person that plays the role of a CPU to execute the program. As you can imagine, this is a pretty boring task that cries out for automation: in this problem, you are asked to write a program that automates the task of the turtle master.

Robot Turtles is played on an $8 \times 8$ board. There is one turtle (marked with the letter T), which always starts out at the bottom-left field, facing right. The board contains empty squares (marked as .), castles made out of rock (C), and castles made out of ice (I). The diamond is marked with a D. The turtle may move only onto empty squares and the square on which the diamond is located.

A turtle program contains $4$ kinds of instructions, marked by a single letter.
 - F The turtle moves one field forward in the direction it is facing. If the turtle faces a castle or the border of the board, a program error occurs.
 - R The turtle turns $90$ degrees to the right (the turtle will just turn and stay on the same field).
 - L The turtle turns $90$ degrees to the left (the turtle will just turn and stay on the same field).
 - X The turtle fires a laser in the direction it is facing. If the square it is facing contains an ice castle, the ice castle will melt and the square will turn into an empty square. Otherwise, a program error occurs. The turtle will not move or change direction. It is a program error to fire the laser at empty squares, rock castles or outside the board.

-----Input-----
The input consists of $9$ lines. The first $8$ lines contains the board, each line representing the squares in a row of the board. The turtle will always be at the bottom-left. There will be exactly $1$ diamond. The $9^{\text {th}}$ line contains the program the turtle master is asked to execute, which contains at least $1$ and at most $60$ instructions.

-----Output-----
Output Diamond! if the entire program can be executed without program error and if the turtle is on the diamond square after the program has been executed. Output Bug! if a program error occurred, or if the turtle does not end up on the diamond square!

-----Examples-----
Sample Input 1:
........
........
........
...CC...
..C.DC..
.C..C...
C.IC....
T.C.....
FLFRXFLFRFLFRF
Sample Output 1:
Diamond!

Sample Input 2:
........
........
........
...CC...
..C.DC..
.C..C...
C.IC....
T.C.....
FLFRFLFRFLFRF
Sample Output 2:
Bug!
user: A coin system $S$ is a finite (nonempty) set of distinct positive integers corresponding to coin values, also called denominations, in a real or imagined monetary system. For example, the coin system in common use in Canada is $\{ 1,5,10,25,100,200\} $, where $1$ corresponds to a $1$-cent coin and $200$ corresponds to a $200$-cent ($2$-dollar) coin. For any coin system $S$, we assume that there is an unlimited supply of coins of each denomination, and we also assume that $S$ contains $1$, since this guarantees that any positive integer can be written as a sum of (possibly repeated) values in $S$.

Cashiers all over the world face (and solve) the following problem: For a given coin system and a positive integer amount owed to a customer, what is the smallest number of coins required to dispense exactly that amount? For example, suppose a cashier in Canada owes a customer $83$ cents. One possible solution is $25+25+10+10+10+1+1+1$, i.e., $8$ coins, but this is not optimal, since the cashier could instead dispense $25+25+25+5+1+1+1$, i.e., $7$ coins (which is optimal in this case). Fortunately, the Canadian coin system has the nice property that the greedy algorithm always yields an optimal solution, as do the coin systems used in most countries. The greedy algorithm involves repeatedly choosing a coin of the largest denomination that is less than or equal to the amount still owed, until the amount owed reaches zero. A coin system for which the greedy algorithm is always optimal is called canonical.

Your challenge is this: Given a coin system $S = \{ c_1, c_2, \ldots , c_ n\} $, determine whether $S$ is canonical or non-canonical. Note that if $S$ is non-canonical then there exists at least one counterexample, i.e., a positive integer $x$ such that the minimum number of coins required to dispense exactly $x$ is less than the number of coins used by the greedy algorithm. An example of a non-canonical coin system is $\{ 1,3,4\} $, for which $6$ is a counterexample, since the greedy algorithm yields $4+1+1$ ($3$ coins), but an optimal solution is $3+3$ ($2$ coins). A useful fact (due to Dexter Kozen and Shmuel Zaks) is that if $S$ is non-canonical, then the smallest counterexample is less than the sum of the two largest denominations.

-----Input-----
Input consists of a single case. The first line contains an integer $n$ $(2 \leq n \leq 100)$, the number of denominations in the coin system. The next line contains the $n$ denominations as space-separated integers $c_1 \ c_2 \ \ldots \ c_ n$, where $c_1 = 1$ and $c_1 < c_2 < \ldots < c_ n \leq 10^6$.

-----Output-----
Output “canonical” if the coin system is canonical, or “non-canonical” if the coin system is non-canonical.

-----Examples-----
Sample Input 1:
4
1 2 4 8
Sample Output 1:
canonical

Sample Input 2:
3
1 5 8
Sample Output 2:
non-canonical
user: You and your friends at the Betting against All Probability Club are visiting a casino where the following game is played.

Each of the $n$ players starts with $k$ lives and puts in a fixed amount of money. In each round of the game, each player flips a biased coin and loses a life if she gets tails. The game ends when only one player remains, in which case this person wins, or ends in a draw if all remaining players lose their last life in the same round. If there is a winner, she wins $n$ times her original bet. In case of a draw, no one wins anything.

Being a BAPC member you quickly realize the casino has an edge here: whenever the game ends in a draw all of the contestants lose the money they bet. You are now wondering what exactly is the probability that this game ends in a draw, so you can figure out how much the casino profits on average.

-----Input-----
 - One line containing two integers, $2\leq n\leq 50$, the number of players, $1\leq k\leq 50$, the number of lives each player has, and a real number $0.1 \leq p \leq 0.9$, with at most two digits after the decimal point, which is the probability the coin lands heads.

-----Output-----
 - Output a single real number: the probability of the game ending in a draw. Your answer should have an absolute error of at most $10^{-6}$.

-----Examples-----
Sample Input 1:
2 2 0.5
Sample Output 1:
0.185185185

Sample Input 2:
2 2 0.8
Sample Output 2:
0.056241426
user: Little Ivica got himself a summer job at a company that produces computer fonts. The branch of the company where Ivica works at specialises in testing computer fonts and Ivica’s team is responsible of testing only lowercase letters of the English alphabet.

The letters are tested so that various sentences using those letters are printed out and then manually (more accurately, visually) checked if everything is arranged properly. Only sentences which contain all 26 lowercase letter of the English alphabet (a–z) are used for testing purposes. These sentences are called test sentences.

You’ve probably already assumed that Ivica’s job is to find test sentences. He has a dictionary which consists of $N$ words and has to calculate how many different test sentences can be made out of those words. Every word from the dictionary can be used only once in the sentence and the order of the words in the sentence is irrelevant (i.e. “uvijek jedem sarmu” and “jedem sarmu uvijek” are equal sentences).

-----Input-----
The first line of input contains the integer $N$ $(1 \leq N \leq 25)$, the number of words in the dictionary. Each of the following $N$ lines contains a single word from the dictionary, its length not exceeding 100. All the words from the dictionary are going to be unique.

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

-----Examples-----
Sample Input 1:
9
the
quick
brown
fox
jumps
over
a
sleazy
dog
Sample Output 1:
2

Sample Input 2:
3
a
b
c
Sample Output 2:
0
user: In the nearby kindergarten they recently made up an attractive game of strength and agility that kids love.

The surface for the game is a large flat area divided into $N \times N$ squares.

The children lay large spongy cues onto the surface. The sides of the cubes are the same length as the sides of the squares. When a cube is put on the surface, its sides are aligned with some square. A cube may be put on another cube too.

Kids enjoy building forts and hiding them, but they always leave behind a huge mess. Because of this, prior to closing the kindergarten, the teachers rearrange all the cubes so that they occupy a rectangle on the surface, with exactly one cube on every square in the rectangle.

In one move, a cube is taken off the top of a square to the top of any other square.

Write a program that, given the state of the surface, calculates the smallest number of moves needed to arrange all cubes into a rectangle.

-----Input-----
The first line contains the integers $N$ and $M$ ($1 \le N \le 100$, $1 \le M \le N^2$), the dimensions of the surface and the number of cubes currently on the surface.

Each of the following $M$ lines contains two integers $R$ and $C$ ($1 \le R, C \le N$), the coordinates of the square that contains the cube.

-----Output-----
Output the smallest number of moves. A solution will always exist.

-----Explanation of Sample Data-----
In the first example, it suffices to move one of the cubes from $(1, 1)$ to $(1, 2)$ or $(2, 1)$.

In the third example, a cube is moved from $(2, 3)$ to $(3, 3)$, from $(4, 2)$ to $(2, 5)$ and from $(4, 4)$ to $(3, 5)$.

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

Sample Input 2:
4 3
2 2
4 4
1 1
Sample Output 2:
2
user: John is a leading organizer of this year’s North America ICPC training camp. The camp lasts several days. On each day, there will be a lecture introducing two problems: one classical problem and one creative problem. Each problem can only be introduced once during the entire camp. Every problem has an integer difficulty level.

John knows that the lecture on each day should not be too overwhelming. Therefore, the sum of the difficulties of the two problems in a single day shall not exceed some fixed value. Also, the two problems on each day should be roughly on the same level. Let $d$’s be the absolute difference between the difficulties of the two problems introduced on any given day. The maximum of all of the $d$s, defined as $D$, should be as small as possible.

If John chooses problems well and arranges them wisely, what is the smallest $D$ he can achieve for the $n$ days of the ICPC training camp?

-----Input-----
The first line of input contains four space-separated integers $n$, $p$, $q$ ($1 \leq n, p, q \leq 2 \cdot 10^5$,

$n \leq \min (p, q)$) and $s$ ($0 \leq s \leq 10^9$), where $n$ is the number of days of the camp, $p$ is the number of classical problems, $q$ is the number of creative problems, and $s$ is the maximum sum of difficulties on any given day.

Each of the next $p$ lines contains an integer $x$ ($0 \le x \le 10^9$). These are difficulties of the $p$ classical problems.

Each of the next $q$ lines contains an integer $y$ ($0 \le y \le 10^9$). These are difficulties of the $q$ creative problems.

-----Output-----
Output a single integer, which is the smallest $D$ John can achieve, or $-1$ if there is no way John can select problems for the $n$ training days.

-----Examples-----
Sample Input 1:
3 4 5 10
3
4
4
9
0
1
5
6
6
Sample Output 1:
2

Sample Input 2:
4 4 4 15
1
5
10
12
1
3
10
14
Sample Output 2:
13
user: -----Input-----
The first line of input contains an integer $1 \le N \le 40$, the degree of the recurrence. The next line of input contains $N+1$ integers $a_0, a_1, \ldots , a_ N$ indicating that the linear recurrence is $x_{t} = a_0 + \sum _{i=1}^N a_ i x_{t-i}$. The next line contains $N$ integers $x_0, \ldots , x_{N-1}$ giving the initial values for the recursion. All the coefficients $a_0, \ldots , a_ N$ and initial values $x_0, \ldots , x_{N-1}$ are integers between $-10^9$ and $10^9$ (inclusive).

The next line contains an integer $1 \le Q \le 10$, the number of queries. Then follow $Q$ lines of queries. Each query consists of two integers $T$, $M$ where $0 \le T \le 10^{18}$ gives the index and $1 \le M \le 10^{9}$ is a moduli.

-----Output-----
For each query $T$, $M$, output a line containing $x_ T \bmod M$.

-----Examples-----
Sample Input 1:
2
0 1 1
0 1
6
1 100000
2 100000
3 100000
4 100000
5 100000
6 100000
Sample Output 1:
1
1
2
3
5
8

Sample Input 2:
2
5 7 9
36713 5637282
4
1 10000
1375 1
3781 23
34683447233 1571385
Sample Output 2:
7282
0
16
299255
user: With modern technology advancement, it is now possible to deliver mail with a robot! There is a neighborhood on a long horizontal road, on which there are $n$ houses numbered $1$ to $n$ from left to right. Every day a mail delivery robot receives a pile of letters with exactly one letter for each house. Due to mechanical restrictions, the robot cannot sort the letters. It always checks the letter on top of the pile, visits the house that should receive that letter and delivers it. The robot repeats this procedure until all the letters are delivered. As a result, each of the $n$ houses is visited by the robot exactly once during the mail delivery of a single day.

The mail delivery robot has a tracking device that records its delivery route. One day the device was broken, and the exact route was lost. However, the technical team managed to recover the moving directions of the robot from the broken device, which are represented as a string consisting of $n-1$ letters. The $i$-th letter of the string is ‘L’ (or ‘R’) if the $(i+1)$-th house visited by the robot is on the left (or right) of the $i$-th house visited. For example, if $n = 4$ and the robot visited the houses in the order of $2, 4, 3, 1$, its moving directions would be “RLL”.

With the moving directions, it may be possible to determine the order in which the robot visited the houses. The technical team has asked you to write a program to do that. There can be multiple orders producing the same moving directions, among which you should find the lexicographically earliest order.

-----Input-----
The input has a single integer $n$ ($2 \leq n \leq 2 \cdot 10^5$) on the first line. The second line has a string of length $n-1$ consisting of letters ‘L’ and ‘R’ giving the moving directions of the robot.

-----Output-----
Output the lexicographically earliest order in which the robot may have visited the houses and delivered the letters according to the moving directions. Consider two different integer sequences of equal length $A = (a_1, a_2, \ldots , a_ k)$ and $B = (b_1, b_2, \ldots , b_ k)$, and let $1 \le i \le k$ be the lowest-numbered index where $a_ i \ne b_ i$. Then $A$ is lexicographically earlier than $B$ if $a_ i < b_ i$; otherwise $B$ is lexicographically earlier than $A$.

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

Sample Input 2:
6
RLLRL
Sample Output 2:
1
4
3
2
6
5