meetforever/GRPO_CL huggingface dataset:
user: Luka started driving international routes with his truck. His biggest problem is the border with Slovenia. The border is a point of entrance into the European Union, so every truck is thoroughly examined. Because of this, Luka always has to wait several hours there. To kill the time, he comes up with various logic and math games.

In one of them, Luka first reads the numbers off of $N$ license plates and writes them down on a piece of paper. Then he tries to find an integer $M$ greater than $1$ such that all integers on the paper give the same remainder when divided by $M$. Luka tries to find as many such integers $M$ as possible.

Write a program that, given Luka’s $N$ integers, determines all such integers $M$.

-----Input-----
The first line contains the integer $N$ ($2 \le N \le 100$), the number of integers on paper. Each of the following $N$ lines contains one integer between $1$ and $1000000000$ (one billion). All these integers will be distinct. The input data will guarantee that at least one integer $M$ will always exist.

-----Output-----
Output all integers $M$ separated by spaces, in any order.

-----Examples-----
Sample Input:
3
6
34
38
Sample Output:
2 4
user: In the computer-generated world Matriks, you work as a consultant for some hip IT companies that need unique slides for their cool office spaces. During the past year, you got a number of job offers at different points in time, measured in milliseconds since the start of the year. For each job you could choose to design a small, medium or humongous slide for the company, which would take $2 \cdot 10^5$, $3 \cdot 10^5$ or $4 \cdot 10^5$ milliseconds respectively, starting immediately after the offer is given (if accepted). During an assignment, you have to decline every other assignment that is offered during that time. When looking back at the year you’re not particularly dissatisfied (who can complain about designing slides?), but perhaps you could have chosen what job offers to accept in a different way to earn even more. You charge your clients $1$ cookie per $10^5$ milliseconds that you work. If you chose what assignments to accept and their lengths optimally, how many cookies could you have earned?

-----Input-----
The first line contains an integer $N$ ($1 \le N \le 10^5$), the number of job offers you got during the year.

The second and final line contains $N$ integers separated with spaces, the times at which you got job offers measured in milliseconds from the start of the year. A year consists of $31556926 \cdot 10^3$ milliseconds. It is guaranteed that each job offer was given at least $4 \cdot 10^5$ milliseconds before the end of the year.

-----Output-----
Output an integer, the maximum number of cookies you could have earned.

-----Explanation of sample 1-----
A possible solution is to take the first, third and fourth jobs and build a humongous slide for each one. This results in $12$ cookies. The time between the job offers is at least $4 \cdot 10^5$ milliseconds, so no assignments overlap.

-----Examples-----
Sample Input 1:
4
10000 400000 500000 900000
Sample Output 1:
12

Sample Input 2:
5
8 10 2 1000000 30556926000
Sample Output 2:
12
user: The mythical planet, El-gă-rizm, comprises $m$ islands surrounded by water. There are $n$ different natural resources to be found at various locations on the islands. Interestingly, each of the $n$ natural resources is present on exactly two islands; thus there are $2n$ total occurrences of resources on El-gă-rizm.

Both the Zax and Xaz beings simultaneously attempt to colonize the islands of El-gă-rizm. The Zax and Xaz mistrust each other. They will share the planet but refuse to share any individual island or any resources on an island. Thus each island is populated by only Zax or Xaz, never both. Collectively, the Zax have access to all natural resources on islands they populate. The Xaz also have access to all natural resources on islands that they populate. In order to survive on El-gă-rizm, a being needs access to all $n$ natural resources.

Can the Zax and Xaz coexist on El-gă-rizm?

-----Input-----
The first line contains the positive integers $m$ and $n$, separated by a space, representing the number of islands and the number of different natural resources, respectively. Following that are $m$ lines. The $i$-th such line specifies the natural resources present on the $i$-th island, as a sequence of integers, separated by spaces. The different natural resources are represented by the integers $1, 2, 3, \ldots , n$. The last value on each of the $m$ lines is always $0$. The value $0$ does not represent a natural resource; it only indicates that there are no more subsequent values on that line.

You may assume that $m$ and $n$ are each no greater than $1000000$.

-----Output-----
The output should be a single line consisting of the string YES if the Zax and Xaz can coexist on El-gă-rizm, or the string NO otherwise.

-----Examples-----
Sample Input:
8 8
0
2 4 0
1 8 0
8 5 0
4 3 7 0
5 2 6 0
1 6 0
7 3 0
Sample Output:
YES
user: Marek loves dancing and he has danced a lot in the last couple of years. He has actually danced so much that he became too good in all of the traditional dances like swing, salsa, ballroom and hip-hop and now all partners he dances with can not keep up with him. Therefore he started to invent his own dances and even tries to convince other people to dance these new dances with him.

Marek got really excited when he heard about the coming wedding of his best friend Miroslav. For a whole month he worked on a special dance for the wedding. The dance was performed by $N$ people and there were $N$ marks on the floor. There was an arrow from each mark to another mark and every mark had exactly one incoming arrow. The arrow could be also pointing back to the same mark.

At the wedding, every person first picked a mark on the floor and no 2 persons picked the same one. Then Marek played some music and every 10 seconds there was a loud signal when all dancers had to move along the arrow on the floor to another mark. The placement of the marks was such that everybody could follow the arrow to the next mark in 10 seconds without any trouble. If an arrow was pointing back to the same mark, the person at the mark just stayed there and maybe did some improvized dance moves on the spot.

A year has passed since Miroslav’s wedding and another wedding is coming up. Marek would like to do a similar dance at this wedding as well. He lost all the drawings he had, but luckily he found two photos from exactly when the dance started and when it ended. Marek also remembers that the signal was triggered $K$ times during the time the song was played, so people moved $K$ times along the arrows.

Given the two photos, can you help Marek reconstruct the arrows on the floor? On the two photos it can be seen for every person to which position he or she moved. Marek therefore numbered the people in the first photo from $1$ to $N$ and then wrote the number of the person whose place they took in the second photo.

Marek’s time is running out, so he is interested in any placement of arrows that could produce the two photos.

-----Input-----
The first line of the input contains two integers $N$ and $K$, $2 \le N \le 10000, 1\le K \le 10^9$. The second line of the input contains $N$ space separated integers $a_1, \dots , a_ N$, denoting that dancer number $i$ ended at the place of dancer number $a_ i$. You may assume that $1 \le a_ i \le N$ for all $i$ and every number between $1$ and $N$ inclusive appears exactly once in the sequence.

-----Output-----
If it is impossible to find a placement of arrows such that the dance performed $K$ times would produce the two photos, print “Impossible”. Otherwise print $N$ numbers on a line, the $i$-th number denoting to which person the arrow leads from person number $i$.

-----Examples-----
Sample Input:
6 2
3 4 5 6 1 2
Sample Output:
5 6 1 2 3 4
user: Filip came across an interesting math problem: He was supposed to find a five-digit number with the following property. If you cross out its first two digits, then place the digit $8$ after the remaining three digits to form a four-digit number and multiply the resulting four-digit number by $4$, you will get the original five-digit number.

After thinking for a while, he found his answer. But now he is wondering, what if he had to find an $m$-digit number, where you have to cross out the first $n$ digits, place a (possibly multi-digit) number $p$ after the remaining $(m-n)$-digit number, then multiply this new number by a number $q$ to get the original number? Please help him!

(Note: Every $m$, $p$ and $q$ needs to start with a non-zero digit.)

-----Input-----
The input consists of a single line containing four numbers: $m$, $n$, $p$, and $q$. You may assume that each number is positive and not larger than $1000000$. You may also assume that $m\geq n$.

-----Output-----
The output consists of a single line. If such an $m$-digit number exists, the line contains the smallest such number. If no such number exists, the line contains the string IMPOSSIBLE.

-----Examples-----
Sample Input:
5 2 8 4
Sample Output:
20512
user: You are given $n$ strings, each a permutation of the first $k$ upper-case letters of the alphabet.

String $s$ is a subsequence of string $t$ if and only if it is possible to delete some (possibly zero) characters from the string $t$ to get the string $s$.

Compute the length of the longest common subsequence of all $n$ strings.

-----Input-----
The first line of input contains two integers $n$ ($1 \le n \le 10^5$) and $k$ ($1 \le k \le 26$), where $n$ is the number of strings, and the strings are all permutations of the first $k$ upper-case letters of the alphabet.

Each of the next $n$ lines contains a single string $t$. It is guaranteed that every $t$ contains each of the first $k$ upper-case letters of the alphabet exactly once.

-----Output-----
Output a single integer, the length of the longest subsequence that appears in all $n$ strings.

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

Sample Input 2:
3 8
HGBDFCAE
ADBGHFCE
HCFGBDAE
Sample Output 2:
3
user: A certain political party with $N$ members wants to develop some brand new politics. In order to do so, the party plans to make a committee for new political development. Clearly, the best politics is developed when all committee members disagree with each other, and when the committee is as large as possible.

In order to figure out which pairs of politicians disagree and which don’t, the party then arranged for every possible pair of politicians to discuss a randomly selected topic. Whenever two politicians couldn’t agree on their assigned topic, this was recorded in the party’s Book of Great Achievements.

Armed with this book, you have now been assigned the task of finding the largest comittee where everyone disagrees. However, finding a large comittee can prove challenging; careful analysis have revealed that for any non-empty group of party members, there is always at least one member of the group who disagrees with (strictly) less than $K$ of the other group members. Obviously, then, the committee can not have more than $K$ members. But is there a choice of committee of this size? Find the size of a largest possible committe such that nobody in that committee agrees.

-----Input-----
The first line contains two integers, $N$ the number of members in the party, and $K$ as described above. Each member is indicated by an integer $i$ between $0$ and $N-1$. After the first line follow $N$ lines, one for each politician $i$, starting with $i = 0$. The line for politician $i$ begins with an integer $D_ i$, and is followed by $D_ i$ integers indicating with which other party members the $i$-th politician disagrees according to the Book of Great Achievements. We always have $0 \leq D_ i < N \leq 50000$, and $1 \leq K \leq 10$.

-----Output-----
Output a single integer, the size of the largest possible comittee.

-----Examples-----
Sample Input:
5 3
2 1 2
3 0 2 3
3 0 1 4
2 1 4
2 2 3
Sample Output:
3
user: Chubby Trung and skinny Hanh are best friend. They are smart and lovely kids, and they are both very good at math and programming. One day, chubby Trung set a puzzle to challenge skinny Hanh. Trung gives a list of $n$ distinct points on a Cartesian coordinate plane to Hanh and asks Hanh to draw a single loop connecting all these points with some conditions:
 - The loop consists of exactly $n$ segments that are parallel to the axes.
 - Each segment’s two ends must be $2$ of the given points. Other than these $2$ points, the segment must not go through any other points from the list.
 - Two consecutive segments in the loop must be perpendicular (i.e. they must form a $90$ degree angle), and they must have exactly one intersection (which is at their common end).
 - The loop must go through all $n$ points. The first and last point of the loop must be the same.
 - The loop must not self-intersect.

In the following examples, the first figure shows a valid loop. The second figure shows an invalid loop because there are segments’ ends at $(2,2)$ which is not in the list. The third one is also invalid because it does not go through all $n$ points. And the last figure is also invalid because there exists $2$ consecutive segments that are not perpendicular.

Your task is to help skinny Hanh determine whether it is possible to create such loop.

-----Input-----
The input starts with an positive integer $t$ – the number of test cases. Then $t$ test cases follow, each has the following format:
 - The first line consists of an integer $n$ – the number of points in the list ($1 \leq n \leq 2 \cdot 10^5$). The sum of $n$ in all test cases does not exceed $10^6$.
 - The $i$-th line of the next $n$ lines contains $2$ integers $x_ i$, $y_ i$ describing the $i$-th point ($0 \leq \lvert x_ i \rvert ,\lvert y_ i \rvert \leq 10^9$). It is guaranteed that no two points have the same coordinates.

-----Output-----
For each test case, if it is possible to draw the loop, print ‘YES’; otherwise, print ‘NO’.

-----Examples-----
Sample Input:
2
6
1 1
1 3
2 2
2 3
3 1
3 2
3
1 1
1 2
2 1
Sample Output:
YES
NO
user: It is Catmas time again, and that means a lot of stress for everyone. In the Kattis family everyone used to buy gifts for everyone else, but this becomes more and more expensive every year, since our family keeps growing. 

To make things easier, we decided that this year each person will buy just one gift. We will then put all the gifts in a big bag and choose an order among ourselves, with all orderings being equally likely. Then, in this order, each person picks a gift from the bag, where each gift is chosen with equal probability. If it is their own gift (which they can easily recognize since everyone in the Kattis family is a creative individual making completely unique Catmas gift wrapping), they put it back in the bag and pick another gift. This can take some time, since it might happen that somebody picks their own gift a few times in a row.

This strategy is not perfect, because the last person might still end up with their own gift. When this happens, everyone has to put their gifts back in the bag, and then we restart the entire process all over from the beginning. Now the question is, how long will we have to wait until the process ends and we can start opening our Catmas gifts? Specifically, given the size $n$ of our family, what is the expected total number of gifts taken out of the bag until the process ends and everyone has gotten their gift?

-----Input-----
The input contains one line with one integer $n$ ($2\leq n\leq 1000$) – the current size of the Kattis family.

-----Output-----
Output the expected total number of gifts taken out of the bag, accurate to within an absolute error of at most $10^{-6}$.

-----Examples-----
Sample Input:
2
Sample Output:
3.000000000
user: Donald is a fence builder. He wants to build a fence that is $N - 1$ meters long. He needs a fence post every meter along the fence, which means he needs $N$ fence posts. Donald has $K$ poles of varying lengths that he wants to use as fence posts. The fence posts must have the same lengths, and be as long as possible. Also, the parts of the poles that are not used for the posts must not be longer than the ones used for the posts. Donald can cut the poles as many times as he wants, and at any position he wants. However, cutting a pole takes time, so he wants to make as few cuts as possible while achieving his other goals.

How many cuts does Donald have to make to get the fence posts for his fence?

-----Input-----
The first line has two space-separated integers, $K$ and $N$. The second line consists of $K$ space-separated integers $p_1$, $p_2$, …, $p_ K$, where $p_ i$ represents the length of the $i$th pole.

-----Output-----
Output the minimum number of cuts needed to build the fence.

-----Limits-----
 - $1 \leq K \leq N \leq 10000$
 - $1 \leq p_ i \leq 10000$

-----Examples-----
Sample Input:
1 2
3
Sample Output:
1
user: A new school year has started. The high school for gifted students of Ho Chi Minh city has $N$ classes numbered from $1$ to $N$. All $N$ classes will go through $M$ studying weeks. The high school has $N$ teachers numbered from $1$ to $N$. At the beginning teacher $i$ is assigned to class $i$.

In order to improve the students adaptability to new knowledge or new teaching methods, the principal has plans to reassign the teachers every few weeks. A reassignment only happens at the beginning of a week (on Monday before classes start) and a week has at most $1$ reassignment. A reassignment is to rotate $K$ teachers $p_1, p_2, p_3, \ldots , p_ K$, described as follows:
 - Teacher $p_ i$ moves to the current class where teacher $p_{i+1}$ is teaching ($1 \leq i < K$).
 - Teacher $p_ K$ moves to the current class where teacher $p_1$ is teaching.

After the reassignment, the teachers will stay at the newly assigned classrom, until he is reassigned again.

The principal continuously add shuffle plans and he also asks questions: “When all the previous reassignment plans comes to effect, which class will teacher $d$ teach on Tuesday of the $x$-th week?” Your task is to help him answer all those questions.

-----Input-----
The first line contains $3$ integers: $N$, $M$ and $Q$ - the number of queries ($1 \leq N,M,Q \leq 10^5$). The next $Q$ lines describes $Q$ queries, each will be in either of the $2$ forms:
 - $0$ $K$ $x$ $p_1$ $p_2$ …$p_ K$ - add a plan to rotate $K$ teachers $p_1, p_2, \ldots p_ K$ on Monday of the $x$-th week ($1 \leq x \leq M$, $2 \leq K \leq 10$, all $K$ values in $p$ are unique).
 - $1$ $d$ $x$ - ask which class is taught by teacher $d$ on Tuesday of the $x$-th week.

It is guaranteed that there are no two queries of type $0$ has the same value $x$.

-----Output-----
For each of the queries type $1$, print the answer in a single line.

-----Clarification for the second sample-----
Initial assignment: 

Class $1$

Class $2$

Class $3$

Week $1$

$1$

$2$

$3$

Week $2$

$1$

$2$

$3$

Week $3$

$1$

$2$

$3$

Week $4$

$1$

$2$

$3$

After rotate $(3, 2)$ on Monday of the $2^\textrm {nd}$ week: 

Class $1$

Class $2$

Class $3$

Week $1$

$1$

$2$

$3$

Week $2$

$1$

$3$

$2$

Week $3$

$1$

$3$

$2$

Week $4$

$1$

$3$

$2$

After rotate $(3,1,2)$ on Monday of the $3^\textrm {rd}$ week: 

Class $1$

Class $2$

Class $3$

Week $1$

$1$

$2$

$3$

Week $2$

$1$

$3$

$2$

Week $3$

$3$

$2$

$1$

Week $4$

$3$

$2$

$1$

-----Examples-----
Sample Input:
3 4 5
1 3 4
0 2 2 3 2
1 3 2
1 2 4
1 1 4
Sample Output:
3
2
3
1
user: While reviewing code recently being checked into the repository, Jim discovered that some employees now and then seemed to copy code right from the Internet into the company code base. This would be a potential disaster as the company then risks getting sued by copyright holders of the original code. The obvious solution, talking to the employees and kindly ask them not to submit any stolen code, seemed to solve the problem. Still, it was decided that a screening process should be introduced to detect newly stolen code. 

The screening would work as follows: Every time new code was checked in, the full contents of the changed files where matched against a repository of known open source code. For each file the longest match, in number of consecutive lines, should be reported.

Comparison is done line by line. Empty lines, and lines only containing space, are ignored during comparison and not counted. Leading and trailing spaces should be ignored completely and consecutive space characters inside the lines are treated as one single space. The comparison is case-sensitive.

-----Input-----
Test data starts with the number $0 \le N \le 100$ of code fragments in the repository. Then follows, for each code fragment, one line containing the file name that the fragment was fetched from and the contents of the fragment on subsequent lines. File names will neither contain whitespace nor be guaranteed to be unique. The name is at most 254 characters long. Each fragment is terminated by ***END*** on a line by its own. This line is not considered being part of the fragment.

After the fragments in the repository have all been listed, comes the actual code snippet to find matches for. This snippet is also terminated by ***END*** on a line by its own.

Lines are guaranteed to be no longer than $254$ characters. No code fragment will be longer than $10000$ lines. Any code and file name lines will only contain the ASCII characters 32-126. The total size of the input file will not exceed $10^6$ characters.

-----Output-----
Output the number of matching consecutive lines (empty lines not counted) in a longest match from the repository, followed by a space-separated list of the file names of each fragments containing a match of this length, given in the order that the matching fragments were presented in the repository description. If no fragments match, write the number 0 on a line of its own.

-----Examples-----
Sample Input:
2
HelloWorld.c
int Main() {
    printf("Hello %d\n",i);
}
***END***
Add.c
int Main() {
  for (int i=0; i<10; i++)
    sum += i;
  printf("SUM %d", sum);
}
***END***
int Main() {
  printf("Hello %d\n",i);
  printf("THE END\n");
}
***END***
Sample Output:
2 HelloWorld.c