meetforever/GRPO_CL huggingface dataset:
user: A National Geographic film crew is visiting the ZOO this week. They are creating a documentary about animal speed and they would like to film one or more cheetahs running at full pace. A solitary running cheetah has been filmed successfully many times. Therefore, the crew is also after a much more spectacular feat: As many as possible cheetahs sprinting on parallel tracks filmed together in one shot.

“No, that is impossible,” said the director. “We cannot squeeze those animals into some sort of a start box, as you probably imagine, and then open the box and make them run all at once. It is clearly too dangerous and unpredictable. No.”

“Then let us make more boxes and open some of them earlier and some of them later,” said one of the filmmakers. “Could that work?”

“And if we open the boxes with the slower cheetahs a bit earlier then after a while the faster ones will be overtaking the slower ones and that would be a great shot,” pointed out another filmmaker. “We would like to see the whole pack rather short with the last animals close the leading ones. As close as possible and at least for a moment.”

It was a long and difficult discussion which ensued, but in the end the circumstances of the experiment were agreed upon.

You are given the start time and the speed of each cheetah. The length of the pack, which is defined as the distance between the first and the last cheetah in the pack, might be different at different moments. Find the minimum length of the pack during the run, where all cheetahs must be running. You may also suppose that the track is so long that the minimum length of the pack happens at least a moment before the first cheetah reaches the finish line.

All start boxes will be so close that you may consider them to be in the same place. The $k$-th cheetah will be released from its start box at the given time $t_ k$. The $k$-th cheetah is expected to run the whole distance at constant speed $v_ k$.

-----Input-----
The first line contains the number of cheetahs $N$ ($1 \leq N \leq 100000$). Next, there are $N$ lines, each line contains two integers $t_ k$, $v_ k$ separated by spaces and representing the start time and velocity of the $k$-th cheetah ($1 \leq k \leq N$). All input values $t_ k$ and $v_ k$ are positive and less than $10^5$.

-----Output-----
Print a single line with one floating point number $L$ specifying the minimum length of the running pack. Your answer should not differ from the correct answer by more than $10^{-2}$. The length of the pack is the distance between the first and the last animal in the pack. The length can be measured at any time $T \geq \max _{k = 1, \ldots , N} t_ k$. We suppose that each cheetah is running at a constant speed for all the time from the start and also at its moment of release from the start box.

-----Examples-----
Sample Input 1:
2
1 1
1 1
Sample Output 1:
0.000

Sample Input 2:
2
1 99999
99999 99999
Sample Output 2:
9999700002.000
user: Sonia is the CEO of the South Western Economic Research Consortium (SWERC). The main asset of SWERC is a group of banks spread out in several countries, which specialize in wire transfers between these countries.

Money is transferred between banks which have transfer agreements. Such agreements settle a fixed fee whenever there is a transfer between these banks. When a client wishes to transfer money to an account in a different bank, the money flows between banks with transfer agreements until it reaches the destination account. For each intermediary transaction, the client will have to pay the corresponding transfer fee.

SWERC’s goal is to provide the cheapest fees for its clients, using only its banks as intermediaries, and profit from its commissions. Things were going quite well until the recent economic crisis. Due to the current situation, governments agreed to impose an extra tax on each wire transfer. Their objective is to both increase the tax income and avoid losing money to tax havens around the world. Hence, the intention is make this extra tax as large as possible (while avoiding too much unrest).

Sonia, being a savvy executive, wants to take advantage of this situation and make sure SWERC provides the cheapest way to transfer money between banks X and Y (their most frequent transfer requests). She will try to lobby the politicians so that the new fee makes this happen. She gathered data about the transfer agreements between banks (including competitors) but has no idea what should be the value of the new fee.Task

Can you help Sonia compute the largest fee so that SWERC can provide cheapest way to transfer money between X and Y?Input

The first line consists of four space-separated integers: $N$ $P$ $X$ $Y$, the number of existing banks, the number of transfer partnerships, and the identifiers of banks X and Y, respectively. The next $P$ lines have three space-separated integers: $a_ i$ $b_ i$$c_ i$, meaning there is a partnership between banks $a_ i$ and $b_ i$ with fee $c_ i$.

A line with an integer $M$, the number of banks owned by SWERC, follows. The next line contains $M$ space-separated integers, the identifiers of these banks. $X$ and $Y$ are always in this list.Constraints

$2 \leq M \leq N \leq 1000$

and

$1 \leq P \leq 10000$

$1 \leq X, Y, a_ i, b_ i \leq N$

and

$X \neq Y$ and $a_ i \neq b_ i$

$1 \leq c_ i \leq 1000000000$Output

The output should be a single integer greater than zero: the largest fee so that SWERC can provide cheapest way to transfer money between $X$ and $Y$. However, if there is no value such that this happens, output Impossible instead. If the fee on each transfer can be infinitely large, output Infinity.Sample Output 1 Explanation

If the extra fee is 4 or more, then SWERC can not provide the cheapest transaction fee. Example: if the fee is 4, SWERC provides a cost of 20, using banks 1, 3, 4, 5 and 6, in this order. However, using bank 2 as an intermediary, we can pay only 19.

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

Sample Input 2:
3 4 1 2
1 2 6
1 3 2
1 2 7
2 3 3
2
1 2
Sample Output 2:
Infinity
user: Gray codes are a classic topic in information theory with a number of practical applications, none of which we are concerned with in this problem. An $n$-bit Gray code is an ordering $(x_1, x_2, \ldots , x_{2^n})$ of all $n$-bit binary strings, with the property that any consecutive pair of strings differ in exactly $1$ bit. More formally, for every $1 \le i < 2^n$, it holds that $d(x_{i}, x_{i+1}) = 1$, where $d(\cdot , \cdot )$ denotes the Hamming distance between two binary strings. For instance, for $n=3$, the sequence $(000, 001, 011, 010, 110, 111, 101, 100)$ is a Gray code. 

While Gray codes are great, they are also a bit, well... gray1. In this problem, we look at a much more colorful variant.

For an integer $n \ge 1$ and set of integers $P \subseteq \{ 1, \ldots , n\} $, we say that an ordering $(x_1, \ldots , x_{2^n})$ of all $n$-bit binary strings is an $n$-bit color code with palette $P$, if for all $1 \le i < 2^n$, it holds that $d(x_ i, x_{i+1}) \in P$, i.e., the number of bits by which any consecutive pair of strings differ is in $P$.

Note that for some palettes, color codes do not exist. For instance, if $n = 6$ and $P = \{ 6\} $, the second string must be the binary negation of the first one, but then the third string must be the negation of the second one, i.e., equal to the first string.

Given $n$ and $P$, can you construct an $n$-bit color code with palette $P$?

-----Input-----
The first line of input consists of two integers $n$ ($1 \le n \le 16$) and $p$ ($1 \le p \le n$). Then follow a line with $p$ distinct integers $s_1, \ldots , s_ p$ ($1 \leq s_ i \leq n$ for each $i$) – the elements of $P$.

-----Output-----
If there is an $n$-bit color code with palette $P$, output $2^n$ lines, containing the elements of such a code, in order. If there are many different codes, any one will be accepted. If no such code exists, output “impossible”.

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

Sample Input 2:
3 1
1
Sample Output 2:
000
001
011
010
110
111
101
100
user: Petra is learning how to add two positive integers in school, but thinks it is a bit too difficult. They are currently working with the standard algorithm for addition, where you first compute the sum of the two units digits, then the sum of the two tens digits, and so on. Whenever the sum of the digits at the same position in the two numbers exceeds 9, a carry digit is added onto the digit of the next higher magnitude. Petra has trouble with the last step – she often forgets to keep track of the carry digit. 

A few weeks ago, she also learnt a simpler method of addition. In this method, you repeatedly add $1$ to one of the numbers and subtract $1$ from the other, until the second one reaches zero. This can of course take a lot of time for large numbers.

Petra now wants to combine the two methods, for fast and error-free addition. Her plan is to first perform the second method one step at a time, until the two numbers would not produce a carry digit when added using the standard algorithm (for positive integers, this always happens eventually). To evaluate the performance of her new method, she has asked you to help her compute the number of steps she must perform of the second method when adding two given integers. Petra may perform the addition by $1$ to either of the two numbers (and subtraction by $1$ from the other).

-----Input-----
The input consists of two lines, each containing a positive integer with at most $10^6$ digits. These are the two integers Petra wants to add.

-----Output-----
Output a single integer, the minimum number of times Petra must add $1$ to one of her numbers (while subtracting $1$ from the other) until they can be added using the standard addition algorithm without any carry digits.

-----Examples-----
Sample Input 1:
10
99
Sample Output 1:
1

Sample Input 2:
90
10
Sample Output 2:
10
user: Young ruler Mirko has declared himself king of dwarves. Upon hearing this, Slavko felt threatened and soon declared himself king of elves! As there cannot be more than one king in the land, they have decided to resolve the issue of power once and for all.

Slavko will, along with the $N$ strongest elves of the kingdom, labeled with numbers from $1$ to $N$, go visit Mirko’s castle. In the castle hall, they will be greeted by the $N$ strongest dwarves sitting in a circle, labelled clockwise with numbers from $1$ to $N$.

Mirko has, upon entering the castle, given a number $A_ i$ to each of Slavko’s elves – the label of the dwarf it will fight against. Unfortunately, he didn’t make sure that each elf should get a unique adversary, and soon a terrible fight broke out.

They have decided to solve the problem in the following way:
 - Slavko will send his elves to the hall one by one, in the order he chooses. The next elf can enter the hall only after the one before him found a place to sit.
 - The elf labeled $k$ will first approach the dwarf labeled $A_ k$. If there isn’t an elf sitting beside the dwarf, he will sit there. Otherwise, he will continue walking, from dwarf to dwarf, clockwise, until he finds an unclaimed dwarf.

Now the $N$ resulting pairs of elves and dwarves compete in armwrestling, and the stronger one always wins.

Slavko is well prepared for this event. He has studied all the fighters and determined the strength of each one. Now he wants to send the elves to the hall in the order which, after they all sit down, will bring the most victories for him.

Help him and calculate the highest number of victories in duels that can be achieved by elves!

-----Input-----
The first line of input contains the integer $N$ ($1 \leq N \leq 5\cdot 10^5$). The second line of input contains $N$ integers $A_ i$ ($1 \leq A_ i \leq N$), the adversaries chosen by Mirko. The third line of input contains $N$ integers $P_ i$ ($1 \leq P_ i \leq 10^9$), the dwarves’ strengths. The fourth line of input contains $N$ integers $V_ i$ ($1 \leq V_ i \leq 10^9$), the elves’ strengths. All strengths from the input will be mutually distinct.

-----Output-----
The first and only line of input must contain the maximum number of victories that can be achieved by elves.

-----Examples-----
Sample Input 1:
3
2 3 3
4 1 10
2 7 3
Sample Output 1:
2

Sample Input 2:
4
3 1 3 3
5 8 7 10
4 1 2 6
Sample Output 2:
1
user: “Never attribute to malice that which is adequately explained by stupidity.” ­–Robert Hanlon.

A well-known trendsetter from a faraway land once needed strings with certain palindromic properties for inspiration. Unable to produce them all herself, she decided to outsource the work. To jog your memory, a palindrome is a string that reads the same forwards and backwards. For example, level and madam are palindromes, while label and maxim are not.

All was going well until the International Cheapskate Palindrome Company (ICPC), the company hired to produce the strings, made a huge blunder and delivered a string that was so horrible, the trendsetter lost all her inspiration. Now, she is suing the ICPC for damages resulting from lost productivity.

The ICPC was supposed to produce a palindrome-rich string­–a string which had many nonempty palindromic substrings. Recall that a substring of a string is a string that can be formed by removing zero or more characters from the start and end of the other string. For example, sig and design are substrings of design, while dig and singed are not.

As a lawyer for the ICPC, you need to show that the ICPC made a good faith effort to produce the required string. To this end, you want to show that while the string $S$ the ICPC produced may not be palindrome rich, it is almost palindrome-rich­–that is, it has many nonempty almost palindromic substrings. A string is called almost palindromic if it is either a palindrome, or can be turned into a palindrome by swapping two characters.

To strengthen your case, you will make $Q$ demonstrations. In the $i^\text {th}$ demonstration, you will consider only the $L_ i^\text {th}$ through $R_ i^\text {th}$ characters of $S$, and show that it is almost palindrome-rich by determining how many nonempty almost palindromic substrings it has.

-----Input-----
The first line of input contains two integers, $N$ ($1 \leq N \leq 200000$) and $Q$ ($1 \leq Q \leq 200000$), the length of $S$ and the number of demonstrations you will make, respectively.

The second line of input contains a string of $N$ lowercase Latin alphabet characters, describing the string $S$.

The next $Q$ lines contain the descriptions of the demonstrations. In particular, the $i^\text {th}$ of these lines contains two integers $L_ i$ and $R_ i$ ($1 \leq L_ i \leq R_ i \leq N$), denoting that for the $i^\text {th}$ demonstration, you will consider only the $L_ i^\text {th}$ through $R_ i^\text {th}$ characters of $S$.

-----Output-----
Output $Q$ lines. The $i^\text {th}$ of these lines should contain a single integer, the number of nonempty almost palindromic substrings in the $i^\text {th}$ demonstration.

-----Examples-----
Sample Input 1:
9 3
beginning
1 5
4 8
1 9
Sample Output 1:
5
11
16

Sample Input 2:
6 1
velvet
1 6
Sample Output 2:
7
user: Mirkos village has only one long street stretching from east to west with M houses. Each house has a unique house number, starting with $1$ and ending with $M$.

Recent storm took out most phone lines so the mayor financed construction of a new one. Mirko is interested in the popularity of this new phone network, so he infiltrated its construction and placed special detectors on some points.

Detector detects any phone call made between two houses, as long as one of them is eastward and the other westward from the point the detector is installed.

At the end of the first month, Mirko removed all detectors and now wonders what is the smallest number of phone calls that could have been made during that month.

-----Input-----
The first line of input contains two integers $N$ ($1 \le N \le 100000$), the number of detectors, and $M$ ($N < M \le 1000000000$), the number of houses in the village. The houses are numbered from $1$ to $M$ (inclusive).

The next $N$ lines contains two numbers each: $P_ i$ ($1 \le P_ i < M$), and $C_ i$ ($1 \le C_ i \le 1000000000$), the position and total number of phone calls detected by detector numbered $i$. We say that a detector is on position $P_ i$ if it is between houses numbered $P_ i$ and $P_ i +1$.

There will never be more than one detector on the same position.

-----Output-----
Output a single integer, the minimal number of phone calls made.

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

Sample Input 2:
2 3
1 23
2 17
Sample Output 2:
23
user: Alob and Bice inherited from their parents a piece of land represented by a simple polygon. There is a canal represented by a straight line. Each brother inherited the lands on one side of the canal. It is possible that the land of each person consists of several disconnected pieces of land. It is also possible that one person does not inherit any land at all.

Alob decided to grow corn fields on his land while Bice decided to grow rice fields. However, they both agree that their fields are symmetrical about the canal.

Your task is to help Alob identify the largest possible area of land to grow corn fields.

Note that a simple polygon is a non-intersecting polygon, and does not have any $3$ consecutive vertices that are collinear.

-----Input-----
 - The first line contains $N$ - the number of vertices representing the simple polygon land ($3 \leq N \leq 100$).
 - In the next $N$ lines, the $i^\textrm {th}$ one contains two integers $x_ i$, $y_ i$ - the coordinate of the $i^\textrm {th}$ vertex ($-1000 \leq x_ i, y_ i \leq 1000$).
 - The last line contains four space-separated integers $x_ a$, $y_ a$, $x_ b$, $y_ b$ representing the coordinates of two distinct points in the straight line canal ($-1000 \leq x_ a, y_ a, x_ b, y_ b \leq 1000$).

-----Output-----
Output a single real number - the largest possible area of land to grow corn fields for Alob. 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 $\frac{|a-b|}{max(1,b)} \leq 10^{-6}$.

-----Sample Clarification-----
These are visualization of the samples below:

First sample:

Second sample:

Third sample:

-----Examples-----
Sample Input 1:
4
0 0
2 0
2 2
0 2
0 -1 0 3
Sample Output 1:
0.0000000000

Sample Input 2:
6
0 1
0 4
3 6
7 5
4 2
7 0
5 7 2 0
Sample Output 2:
9.476048311178
user: Brandon Greg Jr. is an art enthusiast who covered the entirety of the walls of his room with works of art. Brandon’s room is in the shape of a simple polygon (simple means that the polygon does not intersect itself). If Brandon is at point $X$, then he can see all points $Y$ such that the line segment $XY$ (excluding point $Y$) does not intersect any walls. Since Brandon’s happiness is proportional to the amount of art he can see by turning his head, he wants to know the area of the set of points from where he could see all of the walls of his room (and thus see his entire art collection). You may assume that Brandon has a very flexible neck.

-----Input-----
The first line of input contains a single integer $n$ ($3\le n\le 2\cdot 10^5$), representing the number of vertices of Brandon’s room.

Each of the next $n$ lines of input contains two space-separated real numbers $x_ i$ and $y_ i$ ($|x_ i|, |y_ i| \le 10^5$), representing the cartesian coordinates of the $i$-th vertex of Brandon’s room. The coordinates have at most two digits after the decimal place. It is guaranteed that the polygon is simple. The vertices may be given in clockwise or counterclockwise order.

-----Output-----
Output the area of the set of points from where Brandon could see all of the walls of his room. The output will be considered correct if the absolute or relative error is within $10^{-4}$ of the answer (if $a$ is the output and $b$ is the answer, then $a$ will be considered correct if $\frac{|a-b|}{\max \{ 1,b\} } \le 10^{-4}$.

-----Examples-----
Sample Input 1:
5
2 0
1 1
0 2
-2 0
0 -2
Sample Output 1:
8.0000000000

Sample Input 2:
5
0.2 0.00
0 -0.2
0.0 0
-0.2 0
0 0.20
Sample Output 2:
0.0200000000
user: The great Equestria Games are coming!

As you know, the Equestria Games is the biggest sports event in Equestria, where ponies from all around the land come and compete. Of course, such a huge event requires months of planning, lots of money, and, most importantly, kingdom-wide media coverage.

The stadium can be considered as a ring partitioned into $N$ sectors. Due to the intense and bitter rivalry between the fans of different teams, every sector seats only fans of a certain team; the $i^\text {th}$ sector clockwise from the northernmost sector accommodates only the fans of team $A_ i$. Every team has a different distinguishing color, and every fan of course wears a t-shirt with the color of their team.

Now, broadcasting rights of the event are being sold to media companies. Media companies are fussy about getting the perfect coverage, and so each of them wants an exclusive, consecutive range of at least $K$ sectors. Additionally, since they want their coverage to look interesting, they also want that range of sectors to have fans wearing at least $C$ distinct colors.

Obviously, the logistics is a nightmare, so naturally the work has been passed on to you. Can you help determine the maximum number of media companies that can be sold broadcasting rights, if all constraints must be satisfied?

-----Input-----
The first line of input contains three integers, $N$ ($2 \leq N \leq 200000$), $K$ ($1 \leq K \leq N$) and $C$ ($1 \leq C \leq N$), the number of sectors in the stadium, the minimum number of consecutive sectors each media company wants, and the minimum number of distinct colors each media company wants, respectively.

The second line of input contains $N$ integers, $A_1, A_2, \dots , A_ N$ ($1 \leq A_ i \leq 10^9$), the teams the fans assigned to each sector of the stadium support.

-----Output-----
Output a single integer on a line by itself, the maximum number of media companies that can be sold broadcasting rights.

-----Examples-----
Sample Input 1:
9 4 3
1 1 9 9 1 6 6 39 9
Sample Output 1:
2

Sample Input 2:
10 2 2
1 1 1 1 1 2 2 2 2 2
Sample Output 2:
2