meetforever/GRPO_CL huggingface dataset:
user: Let us denote $A_{i,j}$ as the element from matrix $A$ located in the $i$-th row and $j$-th column. We say that the matrix $A$ is cool if this holds:
 - $r, s > 1$
 - $A_{1,1} + A_{r,s} \leq A_{1,s} + A_{r,1}$

where $r$ denotes the number of rows, and $s$ the number of columns of matrix $A$.

Additionally, we say that a matrix is extremely cool if each of its submatrices with at least two rows and two columns is cool.

Your task is to determine the largest number of elements that are contained in an extremely cool submatrix of the given matrix.

-----Input-----
The first line of input contains two integers $R$, $S$ ($2 \leq R, S \leq 1000$) which represent the dimensions of the matrix.

Each of the following $R$ lines contains $S$ integers that represent the elements in the matrix. The elements in the matrix will be integers from the interval $[-10^6, 10^6]$.

-----Output-----
The first and only line of output must contain the maximal number of elements that are contained in an extremely cool submatrix of the matrix from the input. If an extremely cool submatrix doesn’t exist, output $0$.

-----Examples-----
Sample Input 1:
3 3
1 4 10
5 2 6
11 1 3
Sample Output 1:
9

Sample Input 2:
3 3
1 3 1
2 1 2
1 1 1
Sample Output 2:
4
user: Petar is throwing a birthday party and he decided to invite some of the employees of his company where he is the CEO.

Each employee, including Petar, has a unique label from $1$ to $N$, and an accompanying type of jokes they tell $V_ i$. Also, each employee of the company except Petar has exactly one supervisor. Since Petar is the CEO of the company, he has the label $1$ and is directly or indirectly superordinate to all the employees.

At the birthday party, there are certain rules that all people present (including Petar) must follow.
 - At the party, there shouldn’t be two people that tell the same type of jokes.
 - Person $X$ cannot be invited if their direct supervisor is not invited.
 - Person $X$ cannot be invited if the set of jokes the invitees that person $X$ is superior to (directly or indirectly) tell and person $X$ don’t form a set of consecutive numbers.

The numbers in the set are consecutive if the difference between adjacent elements is exactly $1$ when the set is sorted ascendingly. For example, $(3, 1, 2)$ and $(5, 1, 2, 4, 3)$. Petar wants to know how many different sets of jokes he can see at his party with the listed constraints.

-----Input-----
The first line of input contains the integer $N$, ($1 \leq N \leq 10\ 000$). The second line of input contains $N$ integers, the types of jokes person $i$ tells, $V_ i$, ($1 \leq V_ i \leq 100$). Each of the following $N-1$ lines contains two integers $A$ and $B$, ($1 \leq A, B \leq N$), denoting that person $A$ is directly superior to person $B$.

-----Output-----
The first and only line of output must contain the number of different sets of jokes that comply to the previously listed constraints.

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

Sample Input 2:
4
3 4 5 6
1 2
1 3
2 4
Sample Output 2:
3
user: Anica is having peculiar dream. She is dreaming about an infinite board. On that board, an infinite table consisting of infinite rows and infinite columns containing infinite numbers is drawn. Interestingly, each number in the table appears a finite number of times.

The table is of exceptionally regular shape and its values meet the requirements of a simple recursive relation. The first cell of each row contains the ordinal number of that row. $A$ value of a cell that is not in the first column can be calculated by adding up the number in the cell to the left of it and that same number, only written in reverse (in decimal representation).

Formally, if $A(i, j)$ denotes the value in the $i$-th row and the $j$-th column, it holds:
 - $A(i, 1) = i$
 - $A(i, j) = A(i, j - 1) + \mathrm{rev} (A(i, j - 1))$1, for each $j > 1$

1

2

4

8

16

77

154 

2

4

8

16

77

154

605 

3

6

12

33

66

132

363

$\cdots $

4

8

16

77

154

605

1111 

5

10

11

22

44

88

176    

$\vdots $   

$\ddots $

$\qquad $

$\qquad $

$\qquad $

$\qquad $

$\qquad $

$\qquad $

$\qquad $

$\qquad $Table 1: The first few rows and columns of the table 

Anica hasn’t shown too much interest in the board and obliviously passed by it. Behind the board, she noticed a lamp that immediately caught her attention. Anica also caught the lamp’s attention, so the friendly ghost Božo came out of it.

“Anica! If you answer correctly to my $Q$ queries, you will win a package of Dorina wafer or Domaćica cookies, based on your own choice! $I$ wouldn’t want to impose my stance, but in my personal opinion, the Dorina wafer cookies are better. Each query will consist of two integers $A$ and $B$. You must answer how many appearances of numbers from the interval $[A, B]$ there are on the board.”

Unfortunately, Anica couldn’t give an answer to the queries and woke up.

“Ah, I didn’t win the Dorina cookies, but at least I have a task for COCI”, she thought and went along with her business.

-----Input-----
The first line of input contains the integer $Q$ ($1 \leq Q \leq 10^5$), the number of queries. Each of the following $Q$ lines contains two integers $A$ and $B$ ($1 \leq A \leq B \leq 10^{10}$) that represent the interval from the query.

-----Output-----
The $i$-th line of output must contain a single integer—the answer to the $i$-th query.

-----Examples-----
Sample Input 1:
2
1 10
5 8
Sample Output 1:
18
8

Sample Input 2:
3
17 144
121 121
89 98
Sample Output 2:
265
25
10
user: Alice received a cake for her birthday! Her cake can be described by a convex polygon with $n$ vertices. No three vertices are collinear.

Alice will now choose exactly $k$ random vertices ($k{\ge }3$) from her cake and cut a piece, the shape of which is the convex polygon defined by those vertices. Compute the expected area of this piece of cake.

-----Input-----
Each test case will begin with a line with two space-separated integers $n$ and $k$ ($3\! \le \! k\! \le \! n\! \le \! 2500$), where $n$ is the number of vertices of the cake, and $k$ is the number of vertices of the piece that Alice cuts.

Each of the next $n$ lines will contain two space-separated real numbers $x$ and $y$ ($-10.0{\le }x,y{\le }10.0$), where $(x,y)$ is a vertex of the cake. The vertices will be listed in clockwise order. No three vertices will be collinear. All real numbers have at most $6$ digits after the decimal point.

-----Output-----
Output a single real number, which is the expected area of the piece of cake that Alice cuts out. Your answer will be accepted if it is within an absolute error of $10^{-6}$.

-----Examples-----
Sample Input 1:
4 3
0 0
1 1
2 1
1 0
Sample Output 1:
0.50000000

Sample Input 2:
5 5
0 4
4 2
4 1
3 -1
-2 4
Sample Output 2:
12.50000000
user: You are a spy for the Benevolent Agent Protection Center (BAPC) and have recently obtained some top secret information. As you are very excited about this discovery, you want to inform your fellow spies of these newfound documents by sending a message to each of them. One option is to privately message all spies within your organization, but this takes a lot of time. Fortunately, the other spies have networks of their own, so they are able to pass on your message to other spies.

However, there are enemy spies in the network, who will pass along the message to the enemy organization. It is therefore vital that the enemy spies do not receive your message. Luckily, you happen to know who these traitors are, so you can avoid them.

There are two ways for you to send the information to other spies in the network: either by private or by public message. If you use a private message, the receiving spy knows that the message is confidential and they will not tell any other spy. If, on the other hand, you send a spy a public message, the spy will notify all other spies he/she can contact about the message. Since the message is not deemed confidential for them, these spies will in turn contact all spies they can get in contact with and so on (regardless of any messages that they received before). Since you do not want anyone else to know who the enemy spies are, you cannot tell people to contact only specific connections.

Because of the huge size of your spy network, you would like to minimize the number of people you personally need to tell the secret message, while ensuring that no enemy spies receive the message. Can you find out how many spies you need to message?

-----Input-----
The input consists of three parts:
 - The first line contains three integers, $S$, $E$ and $C$. The first integer ($1\le S \le 50{.}000$) denotes the total number of spies in the network (yourself not included), the second integer ($0 \le E \le S$) denotes the total number of enemies in the network and the third integer specifies the total number of connections ($0 \le C \le 10^5$) between spies in the network.
 - Then follow $C$ lines, with two integers $0\leq S_1<S$ and $0\leq S_2<S$ on each line, indicating that there is a connection from spy $S_1$ to spy $S_2$. These connections are not symmetric.
 - Finally, one line with $E$ integers, indicating that these spies are enemies.

You may assume that you can message every spy in the network directly.

-----Output-----
Output a single line with one integer indicating the minimum number of messages you need to send to other spies, counting both private and public messages.

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

Sample Input 2:
4 0 4
0 2
0 1
2 1
2 3

Sample Output 2:
1
user: You are the boss of ACM (Apples, Cherries, and Mangos), an upstanding company with a single goal of world domination.

ACM have provided lots of fruits for the last programming competition for minions in Helsinki. The leftovers should now be shipped to Singapore. There is, however, one constraint: In the case that one box of apples is infested with apple-eating insects and the next one in the line is also a box of apples, the insects might move on to the neighboring box and infect it as well. This constraint is applicable for boxes of cherries and boxes of mangos too.

In order to avoid this, ACM decides that the boxes of fruits are to be sent in such a way that two consecutive boxes contain different types of fruits. The statistics department of ACM wants to know how many ways there are to arrange the shipments of $A$ boxes of apples, $C$ boxes of cherries and $M$ boxes of mangos under this constraint.

Please provide a computer program to compute this for various choices of $A$, $C$, and $M$. Two arrangements are different if there exists $i$ such that the $i$-th box delivered in the two arrangements contain different types of fruits. Since the answer may be very big, output the answer modulo a prime number $10^9+7$.

-----Input-----
The input consists of a single line consisting of three single space separated integers $A$, $C$, and $M$, denoting the total number of boxes of apples, cherries, and mangos you need to ship to Singapore, respectively. All three integers will be between $1$ and $200000$, respectively.

-----Output-----
Output the number of different possible arrangements of delivery, modulo a prime number $10^9+7$. If there is no such order, output $0$.Sample Data Explanation

In the first example, the $6$ possible ways are:
 - Apple, Cherry, Mango, Cherry.
 - Cherry, Apple, Cherry, Mango.
 - Cherry, Apple, Mango, Cherry.
 - Cherry, Mango, Apple, Cherry.
 - Cherry, Mango, Cherry, Apple.
 - Mango, Cherry, Apple, Cherry.

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

Sample Input 2:
2 2 2
Sample Output 2:
30
user: Once upon a time, high up on Mount Olympus, it came to pass that the gods held a competition to see who among them was the best at sorting lists of integers. Eris, the goddess of discord, finds this terribly boring and plans to add some mischief to the mix to make things more fun. She will sabotage the answers of Zeus so that his list of numbers is no longer sorted, which will no doubt be so embarrassing that he becomes furious and starts a minor war. 

Eris must be careful not to be discovered while performing her sabotage, so she decides to only change a single digit in one of the numbers in Zeus’ answer. The resulting number may not have any leading zeros (unless it becomes equal to zero in which case a single zero digit is allowed). Eris can only replace a digit with another digit – adding or removing digits is not allowed.

-----Input-----
The first line of input contains $n$ ($2 \leq n \leq 100$), the length of Zeus’ answer. The second line contains $n$ integers $a_1, a_2, \ldots , a_ n$ ($0 \leq a_1 \le a_2 \le \ldots \le a_ n \leq 10^{15}$), Zeus’ answer.

-----Output-----
If Eris can make the list not be sorted by changing a single digit of one of the numbers, then output $n$ integers $b_1, \ldots , b_ n$, the resulting list of numbers after making the change. Otherwise, output “impossible”. If there are many valid solutions, any one will be accepted.

-----Examples-----
Sample Input 1:
3
2020 2020 2020
Sample Output 1:
2021 2020 2020

Sample Input 2:
2
1 9999999
Sample Output 2:
impossible
user: “A foole and his monie be soone at debate: which after with sorrow repents him too late.” ­–Thomas Tusser.

The real estate bubble has popped, and the country has been thrown into a housing market crisis! You, an aspiring developer, sensed opportunity in the midst of chaos, and have decided to come in and snatch some properties on the cheap, hoping to resell them later for higher value.

Of course, you are a keen buyer, and have done your due research on all of the factors that might affect resale value. One of them is the booming Chinese market. When selling to potential Chinese buyers, it is often important to be culturally sensitive and realize that certain numbers and concepts carry significant meanings in their language, some good and others bad.

In particular,
 - the digit $4$ is to be avoided at all costs, as in Chinese “four” sounds similar to the word “death”;
 - the digit $6$ is lucky, as in Chinese “six” sounds similar to the word “flow”, as in “business flowing smoothly”;
 - the digit $8$ is lucky, as in Chinese “eight” sounds similar to the word “wealth”; however
 - according to yin and yang, one cannot have an extreme abundance of good things; a balance must be struck.

Taking all the above into consideration, you’ve chanced upon a street with a long row of houses, with house numbers from $L$ to $R$ from end to end, inclusive. All of these houses have been foreclosed, and are now up for auction. You decide that you will purchase a house if and only if its house number satisfies all of the following conditions:
 - it contains no digit $4$, and
 - the number of digits that are either $6$ or $8$ is the same as the number of digits that aren’t.

Note that house numbers are written in decimal with no leading zeros.

How many of the houses will you purchase?

-----Input-----
The first and only line of input contains two integers, $L$ and $R$ ($1 \leq L \leq R \leq 10^{200000}$), the inclusive range of house numbers on the street.

-----Output-----
Output a single integer on a line by itself, the number of houses you will purchase, following your rules.

Since this number can be quite large, you should output only the remainder after dividing this number by $10^9+7$.

-----Examples-----
Sample Input 1:
30 70
Sample Output 1:
11

Sample Input 2:
66 69
Sample Output 2:
2
user: Lysias loves his full-rigged ship and takes it out to his island castle as often as possible. On his way back to the castle one day, the engines capable of turning the sails failed. As it is a full-rigged ship, he and his crew are now unable to turn the sails in any direction.

But, as luck would have it, there is a south wind which is blowing them in the right direction…at least approximately. Since they may not get to the castle by wind only, Lysias believes he can use the ocean’s currents to move the ship either east or west temporarily by retracting and lowering the ship’s sails. In that way, the ship can move in two potential directions:
 - Move north by lowering the sails
 - Retract the sails and move in the direction of the current they’re on top of (east/west)

Lysias has dug up an old map of all the currents in this part of the ocean. As he is fond of mathematics, not only does Lysias wonder if it’s possible to get home without fixing the sails; he also wonders how many different routes they can take to get home. Can you help him?

-----Input-----
The first line has three integers: $Y$ and $X$ and $x_{init}$, representing the number of rows and columns of the map, and which column the ship is currently placed at. The ship always starts on the bottom row.

Then follow $Y$ rows, each with $X$ characters each. All characters $C_{x,y}$ is one of ‘~’, ‘#’, ‘@’, ‘>’ and ‘<’. ’~’ represents open sea without any currents, ‘#’ is impassable shallow waters, and ‘>’ and ‘<’ are currents moving the boat to the right and left respectively. ‘@’ represents Lysias’ castle.

-----Output-----
Output all the different distinct paths the ship could take to get back to the castle. Since there may be very many different ways to get back, output the answer modulo $1000003$.

If there are no ways to get to the castle, output “begin repairs”.

-----Limits-----
 - $0 < Y \leq 300$
 - $0 \leq x_{init} < X \leq 50000$
 - If $C_{x,y} = \text {‘>’}$, then $x+1 < X$ and $C_{x+1,y} \notin \{ \text {‘<’}, \verb|`#'|\} $
 - If $C_{x,y} = \text {‘<’}$, then $0 \leq x-1$ and $C_{x-1,y} \notin \{ \text {‘>’}, \verb|`#'|\} $
 - There is exactly one ‘@’ tile on the map
 - The boat will not start on a `#' tile

-----Examples-----
Sample Input 1:
2 2 0
>@
>~
Sample Output 1:
2

Sample Input 2:
3 5 1
>>@<<
>~#~<
>>>>~
Sample Output 2:
4
user: While browsing a math book, Mirko found a strange equation of the form $A=S$. What makes the equation strange is that $A$ and $S$ are not the same. Mirko realized that the left side of the equation should have addition operations between some pairs of digits in $A$.

Write a program that inserts the smallest number of addition operations on the left side to make the equation correct. The numbers in the corrected equation may contain arbitrary amounts of leading zeros.

-----Input-----
The first line contains the equation in the form $A=S$. $A$ and $S$ will both be positive integers without leading zeros. They will be different. $A$ will contain at most $1000$ digits. $S$ will be less than or equal to $5000$. Note: The input data will guarantee that a solution, although not necessarily unique, will always exist.

-----Output-----
Output the corrected equation. If there are multiple solutions, output any of them.

-----Examples-----
Sample Input 1:
143175=120
Sample Output 1:
14+31+75=120

Sample Input 2:
5025=30
Sample Output 2:
5+025=30
user: You are the boss of Wi Know, an upstanding company in information theory, especially in message encryption.

The counter-counter-intelligence branch of your upstanding company managed to intercept a message sent by the counter-intelligence agency of the local Helsinkian government. This message is, of course, of utmost importance, and its content can probably be used for the “greater good” later. The message is a sequence $S$ of $N$ positive integers not greater than $N$, indexed from $1$ to $N$. Let $S_ i$ be the $i^\textrm {th}$ integer of $S$.

As the first step to mine useful information from this message, you first have to find patterns in it. At the moment, the pattern we’re interested in is whether there exists two different integers $A$ and $B$ such that the pattern $A, B, A, B$ appears as a (not necessarily contiguous) subsequence of the original message. That is, whether there exists four indices $1 \le c < d < e < f \le N$ such that $S_ c = S_ e$, $S_ d = S_ f$, and $S_ c \not= S_ d$.

Your task is to find such a pattern, if any, and print both $A$ and $B$. If there are multiple such pairs $(A, B)$, find the lexicographically smallest one. That is, if there are multiple such pairs $(A, B)$, print the one whose $A$ is minimized. If there are still multiple such patterns, print the one whose $B$ is minimized.

-----Input-----
The first line contains a non-negative integer $4 \leq N \leq 400000$, giving the number of integers in $S$. Thereafter follow $N$ lines, the $i^\textrm {th}$ line contains a single integer $1 \le S_ i \le N$.

-----Output-----
If $A \not= B$ exists and the pattern $A, B, A, B$ appears as a subsequence of $S$, you should print two integers $A$ and $B$ on a single line separated by a single space, denoting the lexicographically smallest pair of $(A, B)$ as described in the problem statement. Otherwise, if there is no such pair, you should print a single integer $-1$.

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

Sample Input 2:
8
1
2
3
4
5
6
7
1
Sample Output 2:
-1
user: You have $n$ lights, each with its own button, in a line. Pressing a light’s button will toggle that light’s state; if the light is on, it will turn off, and if the light is off, it will turn on. The lights change at $1$ second timesteps. You can press a button at any time, but it will not take effect until the next timestep. Before each timestep, you may choose to push at most one button (you may also choose to not press any button).

Pushing a button will affect not just the light in question, but all lights down the line. More specifically, if you choose to press the $i^\textrm {th}$ button right before the $k^\textrm {th}$ timestep, then the $(i+m)^\textrm {th}$ light will toggle on the $(k+m)^\textrm {th}$ timestep (with $i+m \le n$). For example, if you press button $5$ just before time $19$, then light $5$ will toggle at time $19$, light $6$ will toggle at time $20$, light $7$ will toggle at time $21$, and so on. If you push a button that will take effect at the same time as its light would have toggled due to an earlier button press, then the two cancel each other out, including subsequent toggles.

Suppose there are three lights, all of which are off at the start. If you press the first button before the first timestep, this will happen in three timesteps:

Now, suppose you press the first button before the first timestep, and then the second button between the first and second timesteps. The button press will cancel out the propagation, and this will happen (note that the propagation will go no further):


 Now, suppose you press the first button before the first timestep, and then the third button between the first and second timesteps. All three lights will be on at the second timestep (but not the third):

You wish to turn on all the lights. What is the earliest time you could possibly see all of the lights turned on? Note that if the lights are all on at time $t$ but not at time $t+1$ due to this propagation, $t$ is still the correct answer.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will consist of a single string $S$ ($1 \le |S| \le 16$). The string $S$ will contain only the characters $1$ and $0$, where $1$ represents that that light is initially on, and $0$ represents that that light is initially off. The first character is light $1$, the next is light $2$, and so on.

-----Output-----
Output a single integer, which is the earliest time at which all of the lights are on.

-----Examples-----
Sample Input 1:
1101
Sample Output 1:
1

Sample Input 2:
1
Sample Output 2:
0
user: You are given an array of $N$ integers. Find a consecutive subsequence of numbers of the length at least $K$ that has the maximal possible average.

-----Input-----
The first line of input contains two integers $N$ ($1 \leq N \leq 3 \cdot 10^5$) and $K$ ($1 \leq K \leq N$). The second line of input contains $N$ integers $a_ i$ ($1 \leq a_ i \leq 10^6$).

-----Output-----
The first and only line of output must contain the maximal possible average. An absolute deviation of $\pm 0.001$ from the official solution is permitted.

-----Examples-----
Sample Input 1:
4 1
1 2 3 4
Sample Output 1:
4.000000

Sample Input 2:
4 2
2 4 3 4
Sample Output 2:
3.666666