meetforever/GRPO_CL huggingface dataset:
user: Yes! You’ve finally been chosen to participate in the "Great Geek Game-show 3000". This is the moment you’ve been waiting for, ever since you puzzled out how to maximise your chances of winning. You will finally be rich, and able to buy all the algorithm books you want! Of course you will have to share the winnings with the other contestants, but since your algorithm is vastly superior to the randomness of all the other numb-skulls you are certain you will be able to keep most of the prize money for yourself, in exchange for telling them how you can all maximise your chances of winning. 

The rules of the game are the following: There is a stage with $N$ boxes, each containing the name of one of the $N$ contestants, and such that each contestant has their name in exactly one box. The contestants enter the stage one at a time, and each one is allowed to peek inside $K$ of the boxes. If they find their own name inside one of these boxes they can get off the stage, and the game continues with the next contestant. If all contestants find their own name, everyone wins. But if one contestant fails to find theirs, everyone loses. After the game has begun, no communication between the contestants is allowed. However you are all allowed to agree upon a strategy before the game begins, and this is where you explain to all the others that the algorithm of everyone choosing $K$ boxes at random is a very bad one, since it gives a chance of winning only equal to $\left(\frac{K}{N}\right)^N$. Instead you suggest the following algorithm:

Assign to each player and each box a unique number in the range $1, \dots , N$. Then each player starts with opening the box with the same number as themselves. The next box the player opens is the box whose number is found inside the first box, then the box whose number is found inside the second box, and so on. The process goes on until the player has opened $K$ boxes, or found their own number.

Now to bring home your point of how superior your algorithm is, you will need to calculate the exact odds of winning if all the contestants follow your directions. Unfortunately, this is the only thing you haven’t figured out yet

-----Input-----
One line with the following integers:

$1 \leq N \leq 10000000$ – the number of contestants.

$1 \leq K \leq N$ – the number of boxes each contestant may check.

-----Output-----
The chances you have of winning if everyone follows your algorithm. The answer should be accurate to an absolute or relative error of at most $10^{-6}$.

-----Examples-----
Sample Input 1:
4 2
Sample Output 1:
0.416667

Sample Input 2:
6 5
Sample Output 2:
0.833333

Sample Input 3:
137 42
Sample Output 3:
0.029351
user: An array in a certain programming language is defined in the following way:
 - The beginning and the end of an array is denoted by an open and closed curly bracket, respectively.
 - Inside the curly braces, there are (possibly even 0) values separated by a comma (there is no comma after the last value in the array).
 - Each value can be a word (an array of lowercase letters of the English alphabet) or another array.
 - Examples of correctly defined arrays are: “{}”, “{a,b,c}”, and “{abc,znj,{novi,niz},pozz}”.

Recently, you’ve noticed that the programming language does not throw an error if you do not place the appropriate number of spaces and new lines before or after the curly braces or the commas. Given the fact that you too mind the values being so “squished” together, you’ve decided to get to work and modify the shape of the array in the following way:
 - Each value that is not an array or denotes the beginning or the end of the array (curly braces) will be in its own line.
 - The commas are “connected” with the value located directly before them and there is a new line after each comma.
 - After the curly bracket denoting the beginning of the array, the indentation of the content increases (shifting the output to the right) for $2$ spaces.
 - Before the curly bracket denoting the end of the array ’}’, the indentation of the content decreases (shifting the output to the left) for $2$ spaces.

-----Input-----
The first line of input contains an array of characters $S$ ($1 \leq |S| \leq 1500$), representing the array from the task.

-----Output-----
The output must consist of the modified version of the array from the task.

-----Examples-----
Sample Input 1:
{abc,ono,sto}
Sample Output 1:
{
  abc,
  ono,
  sto
}

Sample Input 2:
{}
Sample Output 2:
{
}

Sample Input 3:
{znakovi}
Sample Output 3:
{
  znakovi
}
user: Agneta and Beata are visiting the famous Swedish Amusement Park of Infinite Fun and Games. Their favourite attraction is the Infinitely Long Slides. Each slide can be viewed as an infinitely long, completely vertical spiral of radius $1$.

When projected into the (flat) ground, the first slide has $(0, 0)$ as its centre point, and the second one has $(1, 0)$ as its centre point. Furthermore, the start of the slides are at $(1, 0)$ and $(2, 0)$ respectively, and then proceeds counter-clockwise around their respective centres. When Agneta rides her slide (the first one), she does so with vertical speed $1\text { m/s}$, while Beatas slide has a bit higher gradient, so she goes with vertical speed $2\text { m/s}$. However, their angular speeds around the centre is both equal to $1 \text { rad/s}$. Since Beata is a little afraid of the slides, she always waits for Agneta to go first. $W$ seconds after Agneta went, Beata starts her descent.

To make the infinite ride a bit more fun, the amusement park gives a prize if the two riders on the slides manage to pass a ball between them at some point. This prize consists of a lot of chocolate, which is Beata’s favourite treat. To maximize their chances of passing the ball successfully, they want to wait until their Euclidean distance is the smallest possible during the ride. How small will this distance be?

-----Input-----
The first and only line of the input contains the decimal number $0 \le W \le 1000$. The number contains at most $6$ digits after the decimal point.

-----Output-----
Output a single number – the minimal Euclidean distance between Agneta and Beata during their trip. Your answer will be accepted if it is within a relative or absolute error of $10^{-4}$ of the judges’ answer.

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

Sample Input 2:
6.283185
Sample Output 2:
1

Sample Input 3:
42
Sample Output 3:
1.5228490019
user: Alice and Bob have an integer $N$. Alice and Bob are not happy with their integer. Last night they went to a cocktail party and found that another couple had the exact same integer! Because of that they are getting a new integer.

Bob wants to impress the other couple and therefore he thinks their new integer should be strictly larger than $N$.

Alice herself is actually fond of some specific integer $k$. Therefore, Alice thinks that whatever integer they pick, it should be possible to write it as a sum of $k$ distinct powers of $2$.

Bob is also a cheapskate, therefore he wants to spend as little money as possible. Since the cost of an integer is proportional to its size, he wants to get an integer that is as small as possible.

-----Input-----
 - A single line containing two integers $N$ and $k$, with $1 \leq N \leq 10^{18}$ and $1 \leq k \leq 60$.

-----Output-----
Output $M$, the smallest integer larger than $N$ that can be written as the sum of exactly $k$ distinct powers of $2$.

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

Sample Input 2:
12 2
Sample Output 2:
17

Sample Input 3:
1 5
Sample Output 3:
31
user: Illiteracy is a simple puzzle game created by Le Sio. 1 The game consists of a sequence of eight icons; the actual icons are very artistic, but for simplicity, we’ll label the icons using a capital letter A—F. Clicking any icon has a unique effect on the other icons, based on which icon is clicked and that icon’s position in the sequence. Most of the icons rotate the type of other icons. A rotation changes an A to a B, B to C, C to D, D to E, E to F, and F back to A.

Here’s what the icons do when you click on one, based on the type of the icon and its position $x$ in the sequence ($1\leq x \leq 8$):

Type

Effect

A

Rotates the icons immediately to the left and right (at positions $x-1$ and $x+1$), ignoring any icon that doesn’t exist (when $x=1$ or $8$).

B

If the icon is at either end of the sequence, does nothing. Otherwise, changes the icon at position $x+1$ to the same type as the current icon at $x-1$.

C

Rotates the icon at position $9-x$.

D

Rotates all icons between position $x$ and the closest of the sequence’s two ends. (Does nothing if $x$ is one of the two ends, and does not change the icon at position $x$ itself). For example, if $x=3$, the icons at $x=1$ and $2$ would be rotated. If $x=5$, then the icons at positions $6$, $7$, and $8$ would be rotated.

E

If the icon is at either end of the sequence, does nothing. Otherwise, let $y$ be the number of icons between position $x$ and the closest of the sequence’s two ends. Rotates the two icons at positions $x-y$ and $x+y$. For example, if $x=3$, the icons at $x=1$ and $5$ would be rotated. If $x=5$, the icons at positions $8$ and $2$ would be rotated.

F

If $x$ is odd, rotates the icon at position $(x+9)/2$. If $x$ is even, rotates the icon at position $x/2$.

Given a starting sequence of icons and a target sequence, what is the minimal number of clicks required to transform the starting sequence into the target sequence?Examples

For the cases below, we illustrate one possible minimal sequence of clicks to get from the top configuration to the bottom one. The carets indicate which icon on the previous line was clicked to generate the sequence on the following line. The sequence on the left takes $2$ clicks; the sequence on the right takes $4$ clicks.

ABCDEFCD        DCDAFCBA ^      ^   
BCDDEFCD        DCEAACBA ^       ^
BCEDEFCD        DCEAACBC ^      DCEAACCC ^     ECEABCCC


-----Input-----
The input consists of exactly two lines of eight characters each. The first line is the starting icon sequence, and the second is the target sequence. Each character on each line is one of the six capital letters A, B, C, D, E, or F. It will always be possible to reach the target sequence from the given starting sequence using a finite number of clicks.

-----Output-----
Output a single integer, the smallest number of icon clicks needed to get from the starting sequence to the target sequence.

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

Sample Input 2:
DCDAFCBA
ECEABCCC
Sample Output 2:
4

Sample Input 3:
ABCDEFCD
ABCDEFCD
Sample Output 3:
0
user: After seeing all the beautiful landscaping around Cossin, Yraglac is inspired to work on his own gardening. Since food is such a scarce commodity on Mars, he decides it would be nice to plant an agricultural crop: Martian rice. The unique thing about Martian rice is that it will only grow in flat areas where water can pool, which makes it important to terrace the garden carefully. Given a height profile of Yraglac’s garden, can you help him determine how much of the land can grow rice on?

Yraglac’s garden is divided into a regular grid of perfectly square $1 \textrm{ m } \times 1 \textrm{ m}$ cells, and he provides you a map indicating the exact height of the land within each cell. The entire garden is surrounded by a wall that is higher than the highest cell. In our model of the Mars world, rain water (real or artificial) can flow from a cell to any of its four adjacent cells (north, east, south, or west), provided the adjacent cell’s height is lower than or equal to the cell’s height. A cell is said to be able to collect or pool water if any water landing in the cell cannot flow to another cell of lower height, either directly or through any of its neighbouring cells. Arrows in the diagram below illustrate possible directions of water flow between cells, and circles indicate the four cells that can collect water.

-----Input-----
The input begins with two integers, $x \; y$, which indicate the dimensions of Yraglac’s garden, in metres ($1 \le x, y \le 500$). Then following $y$ lines containing $x$ integers, $h_{ij}$, each which indicate the heights of each cell in Yraglac’s garden ($0 \le h_{ij} \le 999$).

-----Output-----
Output the number of square metres of land that Yraglac can grow his rice crop on.

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

Sample Input 2:
7 2
0 4 1 4 2 4 3
0 4 1 4 2 4 3
Sample Output 2:
8

Sample Input 3:
5 3
1 1 1 1 1
3 3 3 3 3
5 5 5 5 5
Sample Output 3:
5
user: After Curiosity discovered not just water on Mars, but also an aggressive, bloodthirsty bunch of aliens, the Louvain-la-Neuve municipal government decided to take precautionary measures; they built shelters in order to shelter everyone in the city in the event of an extraterrestial attack.

Several alien-proof shelters have been erected throughout the city, where citizens can weather an alien invasion. However, due to municipal regulations and local building codes the shelters are limited in size. This makes it necessary for the government to assign every citizen a shelter to calmly direct themselves towards in the rare event of a fleet of UFOs blotting out the sun. Conditional on no shelter being assigned more people than it can fit, it is of the utmost importance that the time it takes until everyone has arrived at a shelter is minimized.

We model Louvain-la-Neuve as a network of $n$ locations at which people live, connected by $m$ bidirectional roads. Located at $s$ points throughout the city are the shelters, each with a given maximum capacity. What is the minimum amount of time it takes for everyone to arrive at a shelter, when we assign people to shelters optimally?

The Louvain-la-Neuve municipal government has made sure that there is enough shelter capacity for its citizens and all shelters can be reached from any location, i.e. it is always possible to shelter everyone in some way.

-----Input-----
 - On the first line are three integers, the number of locations $1 \leq n \leq 10^5$, roads $0 \leq m \leq 2\cdot 10^5$, and shelters $1 \leq s \leq 10$.
 - Then follows a line with $n$ integers $0 \leq p_ i \leq 10^9$, indicating the the number of people living at location $1 \leq i \leq n$.
 - Then follow $m$ lines containing three integers $1 \leq u, v \leq n$ and $1 \leq w \leq 10^9$ indicating that there is a bidirectional road connecting $u$ and $v$ that takes $w$ time to traverse. For any two locations there is at most one road connecting them directly, and no road connects a location to itself.
 - Finally follow $s$ lines with two integers $1 \leq s_ i \leq n$ and $1 \leq c_ i \leq 10^9$, indicating that there is a shelter with capacity $c_ i$ at location $s_ i$.

-----Output-----
Print the minimum amount of time it takes to shelter everyone.

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

Sample Input 2:
4 5 2
2 0 0 2
1 2 6
1 3 2
2 3 3
3 4 4
4 2 6
3 2
2 2
Sample Output 2:
5

Sample Input 3:
7 8 3
0 1 1 1 1 0 2
1 2 1
2 3 1
3 1 1
4 6 5
4 3 1
6 7 10
7 5 3
5 6 3
6 5
1 1
2 1
Sample Output 3:
6
user: A set of gears is installed on the plane. You are given the center coordinate and radius of each gear, which are all integer-valued. For a given source and target gear, indicate what happens to the target gear if you attempt to turn the source gear. Possibilities are:
 - The source gear cannot move, because it would drive some gear in the arrangement to turn in both directions.
 - The source gear can move, but it is not connected to the target gear.
 - The source gear turns the target gear, at a certain ratio

If the source gear cannot move, give this result, even if the source and target gears are not connected.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input contains a single integer $n$ ($1 \le n \le 1000$), the total number of gears. Following this will be $n$ lines, one per gear, containing three integers: the $x, y$ ($-10000 \le x, y \le 10000$) and $r$ ($1 \le r \le 10000$) values for the gear, where $(x,y)$ is the position of the axle of the gear, and $r$ is its radius. Assume that the teeth of the gears are properly designed, and accounted for in the radius, so that any gear will mesh with any other gear if (and only if) they are tangent to each other. The gears will never overlap. The source gear is the first gear, the target gear is the last gear specified.

-----Output-----
Output a single line, with the following content, based on the result:
 - -1 if the source gear cannot move.
 - 0 if the source gear can move but is not connected to the target.
 - a b if the source gear moves the target gear, where $a$ and $b$ are two space-separated integers, and $a:b$ is the ratio of source gear revolutions to target gear revolutions reduced to its lowest form (i.e. they have no common factor other than $1$).
 - $a$ is always positive.
 - If the target turns in the same direction as the source, $b$ is positive.
 - If the target turns in the opposite direction as the source, $b$ is negative.

-----Examples-----
Sample Input 1:
2
0 0 100
0 300 200
Sample Output 1:
2 -1

Sample Input 2:
2
0 0 100
0 300 100
Sample Output 2:
0

Sample Input 3:
16
10 10 5
20 10 5
30 10 5
40 10 5
10 20 5
20 20 5
30 20 5
40 20 5
10 30 5
20 30 5
30 30 5
40 30 5
10 40 5
20 40 5
30 40 5
40 40 5
Sample Output 3:
1 1
user: A scientist, E. Collie, is going to do some experiments with bacteria. Right now, she has one bacterium. She already knows that this species of bacteria doubles itself every hour. Hence, after one hour there will be $2$ bacteria.

E. Collie will do one experiment every hour, for $n$ consecutive hours. She starts the first experiment exactly one hour after the first bacterium starts growing. In experiment $i$ she will need $b_ i$ bacteria.

How many bacteria will be left directly after starting the last experiment? If at any point there are not enough bacteria to do the experiment, print “error”.

Since the answer may be very large, please print it modulo $10^9+7$.

-----Input-----
The input consists of two lines.
 - The first line contains an integer $1\leq n\leq 10^5$, the number of experiments.
 - The second line contains $n$ integers $b_1,\dots ,b_ n$, where $0\leq b_ i\leq 2^{60}$ is the number of bacteria used in the $i$th experiment.

-----Output-----
Output a single line containing the number of bacteria that remains after doing all the experiments, or “error”.

-----Examples-----
Sample Input 1:
3
0 0 0
Sample Output 1:
8

Sample Input 2:
5
1 1 1 1 1
Sample Output 2:
1

Sample Input 3:
5
0 2 2 4 0
Sample Output 3:
0
user: An artist begins with a roll of ribbon, one inch wide. She clips it into pieces of various integral lengths, then aligns them with the bottom of a frame, rising vertically in columns, to form a mountain scene. A mountain scene must be uneven; if all columns are the same height, it’s a plain scene, not a mountain scene! It is possible that she may not use all of the ribbon.

If our artist has $4$ inches of ribbon and a $2 \times 2$ inch frame, she could form these scenes:

She would not form these scenes, because they’re plains, not mountains!

Given the length of the ribbon and the width and height of the frame, all in inches, how many different mountain scenes can she create? Two scenes are different if the regions covered by ribbon are different. There’s no point in putting more than one piece of ribbon in any column.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The input will consist of a single line with three space-separated integers $n$, $w$ and $h$, where $n$ ($0 \le n \le 10000$) is the length of the ribbon in inches, $w$ ($1 \le w \le 100$) is the width and $h$ ($1 \le h \le 100$) is the height of the frame, both in inches.

-----Output-----
Output a single integer, indicating the total number of mountain scenes our artist could possibly make, modulo $10^9 + 7$.

-----Examples-----
Sample Input 1:
25 5 5
Sample Output 1:
7770

Sample Input 2:
15 5 5
Sample Output 2:
6050

Sample Input 3:
10 10 1
Sample Output 3:
1022
user: Ever wonder what will happen when you meet a person who uses a different base system to talk about numbers? For example, when you hear someone saying this mysterious number 1010110001 (which is in binary), can you quickly convert it to its decimal form, 689?

In Harkerland, everyone uses the octal numeral system (base $8$) when talking about numbers. Therefore, new immigrants often have difficulties parsing numbers quickly. One of them is Alex. Alex used to live in a place where the hexadecimal numeral system (base $16$) is adopted. In order to adapt to the Harkerland lifestyle as soon as possible, he trains hard to perform fast octal-to-hexadecimal conversion. Before Alex’s training finishes, he is going to rely on a program for the base conversion. Help Alex write such a program.

Use capital letters A, B, $\dots $, F to represent the values $10, 11, \ldots , 15$ in hexadecimal.

-----Input-----
The only line of input consists of a non-negative integer, written in base $8$. The given integer will have no leading zeroes and will be strictly smaller than $8^{200000}$.

-----Output-----
Output the same integer, written in base $16$. The output must not include any extra leading zeroes.

-----Explanation for sample data-----
For Sample Input 1, $4444_8 = 2340_{10} = 924_{16}$.

-----Examples-----
Sample Input 1:
4444
Sample Output 1:
924

Sample Input 2:
20
Sample Output 2:
10

Sample Input 3:
3211
Sample Output 3:
689
user: You have recently acquired a new job at the Bank for Acquiring Peculiar Currencies. Here people can make payments, and deposit or withdraw money in all kinds of strange currencies. At your first day on the job you help a customer from Nijmegia, a small insignificant country famous for its enormous coins with values equal to powers of $10$, that is, $1, 10, 100, 1000$, etc. This customer wants to make a rather large payment, and you are not looking forward to the prospect of carrying all those coins to and from the vault.

You therefore decide to think things over first. You have an enormous supply of Nijmegian coins in reserve, as does the customer (most citizens from Nijmegia are extremely strong). You now want to minimize the total number of coins that are exchanged, in either direction, to make the exact payment the customer has to make.

For example, if the customer wants to pay $83$ coins there are many ways to make the exchange. Here are three possibilities:Option 1.

The customer pays $8$ coins of value $10$, and $3$ coins of value $1$. This requires exchanging $8 + 3 = 11$ coins.Option 2.

The customer pays a coin of value $100$, and you return a coin of value $10$, and $7$ coins of value $1$. This requires exchanging $1 + 1 + 7 = 9$ coins.Option 3.

The customer pays a coin of value $100$, and $3$ coins of value $1$. You return $2$ coins of value $10$. This requires exchanging $1 + 3 + 2 = 6$ coins.

It turns out the last way of doing it requires the least coins possible.

-----Input-----
 - A single integer $0\leq n < 10^{1000}$, the amount the customer from Nijmegia has to pay.

-----Output-----
 - Output the minimum number of coins that have to be exchanged to make the required payment.

-----Examples-----
Sample Input 1:
83
Sample Output 1:
6

Sample Input 2:
13
Sample Output 2:
4

Sample Input 3:
0
Sample Output 3:
0