meetforever/GRPO_CL huggingface dataset:
user: You are given an $r\! \times \! c$ grid. Each cell of this grid is filled with a number between $1$ and $r{\cdot }c$ inclusive, and each cell’s number is distinct.

Define a grid of numbers to be monotonic if each row and column is either increasing or decreasing (this can be different for each row or column).

Define a subgrid of the grid as follows: First choose some nonempty subset of the rows and columns. Next, take elements that lie in both the chosen rows and columns in the same order.

There are $(2^r{-}1)(2^c{-}1)$ nonempty subgrids of the given grid. Of these subgrids, count how many are monotonic.

Consider this grid:$1$  $2$  $5$

$7$  $6$  $4$

$9$  $8$  $3$


There are nine $1{\times }1$ subgrids, nine $1{\times }2$’s, three $1{\times }3$’s, nine $2{\times }1$’s, nine $2{\times }2$’s, three $2{\times }3$’s, three $3{\times }1$’s, three $3{\times }2$’s, and one $3{\times }3$. They are all monotonic, for $9{+}9{+}3{+}9{+}9{+}3{+}3{+}3{+}1=49$ monotonic subgrids.

-----Input-----
Each test case will begin with a line with two space-separated integers $r$ and $c$ ($1\! \le \! r,c\! \le \! 20$), which are the dimensions of the grid.

Each of the next $r$ lines will contain $c$ space-separated integers $x$ ($1\! \le \! x\! \le \! r{\cdot }c$, all $x$’s are unique). This is the grid.

-----Output-----
Output a single integer, which is the number of monotonic subgrids in the given grid.

-----Examples-----
Sample Input:
3 3
1 2 5
7 6 4
9 8 3
Sample Output:
49
user: The Imitation Game is a 2014 film based in 1939 during World War II. It follows the newly created British intelligence agency MI6 as they employ Cambridge mathematics alumnus Alan Turing to crack the German Enigma machine. At the time, cryptographers everywhere believed it to be uncrackable. By the end of the film, Turing and his team were able to successfully crack the Enigma, ultimately winning the war for the allies.


If you have seen the film, you would know that, brilliant as he is, Alan Turing didn’t crack the code on his own. He employed help from his group of carefully selected colleagues. But how exactly did he select them? He placed crosswords puzzles all over Great Britain in the hopes that other brilliant minds would be able to solve them to qualify as candidates for his challenging project.


Those who solved his crosswords were given a special test to further evaluate their candidacy. Turing’s crosswords were extremely befuddling, but you are smart enough to figure out the words from the clues alone. However, you’re not really a spatial thinker, so you need some help positioning them onto the grid. Thanks to the help of modern day computing, perhaps you too can consider yourself worthy to become part of his famous team.


Given an empty crossword puzzle and a scrambled list of all of the solutions to the clues, your task is to position them appropriately on the grid. Like all typical crosswords, words are only valid when read left-right or up-down. Words must start/end either on the edge of the grid or beside a void space.

-----Input-----
The first line of input consists of two space-separated integers, $R$ and $C$ ($1 \leq R, C \leq 21$), specifying the number of rows and columns in the crossword grid.


$R$ line follow, each of which consists of $C$ characters, specifying the grid for the unsolved crossword. Within this grid, a “#" character represents a void space (i.e. a space where no letters may be placed) and a “." character represents an empty space, where a letter should be placed.


The next line consists of a single integer $N$ ($1 \leq N \leq 200$).


$N$ lines follow, each of which consists of a single string of at least length $2$, consisting of only uppercase letters from “A" to “Z". These lines, given in no particular order, specify every word of the crossword solution that must be placed either horizontally or vertically on the grid.

-----Output-----
Print an $R$ by $C$ grid consisting of the solved crossword.

 You may assume that there will be exactly one solution in every test case.


-----Examples-----
Sample Input:
1 15
##.........####
1
CROSSWORD
Sample Output:
##CROSSWORD####
user: Mirko received an $N \times N$ table for his birthday, where a non-negative integer is written in each field of the table. Unfortunately, the written numbers are too large for Mirko’s taste, so he will place $K$ dominoes on top of the table that will cover the fields that are too large. More precisely, Mirko places the dominoes according to the following rules:
 - each domino covers two fields of the table that are adjacent in a row or in a column,
 - the dominoes do not overlap (but can touch),
 - the sum of all visible (uncovered) fields needs to be as small as possible.

It is your task to determine the required minimal sum of visible fields. The test data will be such that it will always be possible to place $K$ dominoes without overlapping.

-----Input-----
The first line of input contains the integers $N$ ($1 \leq N \leq 2000$), the dimensions of the table, and $K$ ($1 \leq K \leq 8$), the number of dominoes. Each of the following $N$ lines contains $N$ integers from the interval $[0, 1000]$. These $N \times N$ numbers describe Mirko’s table.

-----Output-----
The first and only line of output must contain the minimal sum of visible fields after covering the table with dominoes.

-----Examples-----
Sample Input:
3 1
2 7 6
9 5 1
4 3 8
Sample Output:
31
user: One nice summer day while Mirko was drinking lemonade in his room...

“Big brother!”, yells Stanko.

“I wonder sometimes which of the two of us is the big one. What is it?”, Mirko asked.

“Listen carefully! In the backyard I have $N$ pebbles arranged in a circle. Some of the pebbles are black, some are white. I will do the following: between any two neighbouring pebbles of the same colour I will insert a black pebble, and between any two neighbouring pebbles of different colours I will insert a white pebble. At that point there will be $2N$ pebbles in the circle, so I will remove the starting $N$ pebbles so that only the newly added $N$ pebbles remain. And all this I intend to do exactly $K$ times. And then you are to determine my starting circle.”, said Stanko long-windedly.

“Ha! I shall not fall prey to your trickery! I can see that it is not necessarily possible to know exactly what the starting circle was, but I can count the number of distinct starting circles that give the same result as your circle after exactly $K$ of those weird transformations of yours”, answered Mirko.

You are given the configuration of the circle before Stanko performed the transformation described above $K$ times.

Write a program that determines the number of distinct starting circles that give the same circle after $K$ transformations as Stanko’s original circle does after $K$ transformations.

Two configurations of pebbles are considered to be the same circle if one can be gotten from the other by rotating it any number of positions. For example BBW and BWB is the same circle whereas BBWWBW and WWBBWB are not.

-----Input-----
The first line of input contains two integers $N$ and $K$, $3 \le N \le 100$, $1 \le K \le 10$, where $N$ is the number of pebbles in the circle and $K$ is the number of transformations made by Stanko. The second line contains exactly $N$ characters ‘B’ or ‘W’ representing Stanko’s original circle.

-----Output-----
Output the number of possible distinct starting circles on a single line.

-----Examples-----
Sample Input:
3 1
BBW
Sample Output:
2
user: Young Mr. Potato is opening two new stores where he will, you guessed it, sell potatoes. Mr. Potato gets his potatoes from $N$ farmers. Each farmer offers exactly $a_ i$ potatoes per bag for a total price of $c_ i$. Mr. Potato is going to buy all bags of potatoes from all farmers and place the bags in his two stores.

Let’s denote the average potato price in the first store with $P_1$, and the average potato price in the second store with $P_2$ . The average potato price in a store is equal to the ratio of the price and the total number of potatoes in the store. Taking into account logistical difficulties and the amount of potatoes in the stores, he wants the product of the average prices of potatoes in the stores to be minimal. In other words, he wants the product of $P_1$ and $P_2$ to be minimal.

After Mr. Potato settles on a division of bags in the stores, at least one store must have exactly $L$ bags.

-----Input-----
The first line of input contains two integers $N$ and $L$ ($2 \leq N \leq 100$, $1 \leq L < N$), the number of potato bags and the number of potato bags in at least one store. The second line of input contains $N$ integers $a_ i$ ($1 \leq a_ i \leq 100$), separated by space. The third line of input contains $N$ integers $c_ i$ ($1 \leq c_ i \leq 1000000$), separated by space. The sum of all $a_ i$ will be at most $500$.

-----Output-----
The first and only line of output must contain the minimal product of $P_1$ and $P_2$ from the task. An answer correct up to three decimal places will be accepted.

-----Examples-----
Sample Input:
3 1
3 2 1
1 2 3
Sample Output:
0.556
user: When Farmer Oskar doesn’t watch his cows closely enough, they tend to wander off into the forest to hunt for horse spies. To catch an enemy agent horse, the cows lure it into a cow trail and set up fences at the endpoints of the trail, which are $L$ meters apart. Two previously hidden cows then reveal themselves, having concealed themselves in nearby bushes – and the chase is on! The cows are located at positions $A$ and $B$ meters from the left endpoint.

One particular horse (positioned $P$ meters from the left endpoint) who they attempted to catch remembered to bring her walkie-talkie and have now called for backup. The backup will arrive to rescue her very soon, so she wonders for how long she can hold the cows off.

Horse chasing happens in $1$ minute steps. During each minute, the following steps happen in order:
 - The cows choose to move either $0$ or $1$ meters, in any direction.
 - The horse jumps either $0$, $1$, or $2$ meters in any direction. However, if a cow occupies the same position as the horse, she can only jump a single meter since she now must jump over the cow.
 - If there is a cow in the same position as the horse, the cows capture her.

How long will it take for the horse to be captured, assuming both she and the cows move optimally?

-----Input-----
The first and only line of input contains four integers $1 \le L \le 1000$, $0 \le A \le L$, $0 \le B \le L$, $0 \le P \le L$ separated by spaces, the meanings of which are described in the statement.

The numbers $A$, $B$ and $P$ are all distinct.

-----Output-----
A single integer, the number of minutes before the cows can catch the horse no matter how the horse moves.

-----Examples-----
Sample Input:
5 4 3 2
Sample Output:
3
user: Victor works for Alberta Chemicals Monitoring (ACM). ACM is a company that analyses raw environmental data related to chemicals used in oil sand and other industries in Alberta, and produces some reports for environmental watchdogs.

Victor is in charge of a multi-processor cluster in ACM. Each processor is connected to a dedicated special purpose output generation unit (OGU). This cluster receives several raw data streams from field sensors and assigns each stream to a processor. Each processor performs some real time processing on a data stream and immediately after its termination, produces a report using its OGU.

Each stream has an integer starting time $s$, an integer duration $d$ and a priority $p$. This stream is active in the interval $[s, s+d)$ (right-open interval). The report of each stream must be produced immediately after its termination; otherwise, it will be useless. An OGU creates a report extremely fast, so you can assume that an OGU produces this report instantly.

In the past, at any instance of time, the number of data streams were not more than the number of processors and OGUs. So, Victor could process all data streams. Unfortunately, recently, in a suspicious power surge, all OGUs burnt out. Victor was able to salvage one OGU by using parts from the other OGUs. Now, he can no longer produce a report for all data streams and needs to choose a subset of them based on the priorities assigned to them. To handle access to this OGU, Victor restructured the cluster architecture as follows. When a stream starts, the system either admits or rejects it. If it admits a stream, the unique identifier of the processor assigned to this stream is pushed onto the stack. Only a processor having its identifier on top of the stack can use the OGU to produce its report. After production of the report, the processor identifier is popped from the stack. It should be noted that if some streams start at the same time, he can push their processor identifier in any order of his choice. Now, Victor needs your help to choose a subset of streams such that their reports can be generated with this single OGU. The total priority of the streams in the chosen subset should be maximized.

-----Input-----
The input consists of a single test case. The first line contains an integer $n$, where $n$ ($1 \le n \le 5000$) is the number of data streams. Each of the next $n$ lines contains three integers $s_ i$, $d_ i$, $p_ i$ ($1 \le s_ i,d_ i \le 10^9$, $0 \le p_ i \le 100000$) describing one data stream, where $s_ i$ is its start time, $d_ i$ is the duration of the stream, and $p_ i$ is its priority. Note that the cluster has at least $5000$ processors.

-----Output-----
Display the maximum total priority of a subset of streams such that their reports can be generated with the architecture described above using a single OGU.

-----Examples-----
Sample Input:
4
1 3 6
2 5 8
3 3 5
5 3 6
Sample Output:
13
user: “Weeeee,” says the fox after breaking in to the roost and killing all the chickens. Then he realizes that he has killed way more chickens then he can eat right now and hence he decides to hide them in already existing hiding spots nearby. The fox can only carry two hens at a time and every hiding spot can fit at most one. As he is already tired after all the killing, he wants you to compute the minimum distance he has to walk to hide all of the dead chickens.

-----Input-----
The first line of input gives the position of the roost as two real numbers $x$ and $y$ on a single line. The next line contains an integer $1 \leq N \leq 20$, giving the number of hiding spots and dead chickens. Then follow $N$ lines, each line consisting of two real numbers giving the coordinates of a hiding spot. All coordinate values are between $0$ and $1000$ and given with $6$-digit precision.

-----Output-----
A single line with a floating point number, the minimum distance the fox has to walk to hide all the chickens. Any answer with an error of at most $10^{-6}$ will be judged as correct.

-----Examples-----
Sample Input:
10.000000 20.123456
1
13.141593 20.123456
Sample Output:
3.141593
user: Optimistan is a strange country. It is situated on an island with a huge desert in the middle, so most people live in port towns along the coast. As the name suggests, people of Optimistan (also called Optimists) like to optimise everything, so they only built roads necessary to connect all port towns together and not a single extra road. That means that there is only one way to get from one port town to another without visiting the same place twice.

The government installed multi-directional distance signs in $1$-kilometre intervals on one side of the road, to provide important information to drivers. Thus whenever you go from one port town to another, you pass the first sign at the port town and then one each kilometre. Every distance sign contains the shortest distances to all port towns, each written on a separate small sign directed towards the goal town.

The signs also serve another important function: to guide drivers on intersections. This means that distance of each intersection from every port town is an integer number of kilometres.

You bought a tourist guide of Optimistan which does not have a map of the country, but it contains a huge table with the shortest distances between all pairs of port towns. You quickly calculated the average shortest distance between all pairs of port towns, but then you started wondering: if the signs also contained shortest distances to all other signs, what would be the average number written on a sign? Could this be calculated just from the distance table in the tourist guide?

-----Input-----
The input consists of:
 - one line with an integer $n$ ($2 \le n \le 500$), the number of ports;
 - $n-1$ lines, the $i$th of which contains $n-i$ integers. The $j$th integer on the $i$th line denotes the distance between port $i$ and port $i+j$ in kilometres. Each distance is between $1$ and $10^6$ (inclusive).

You can assume that the distances correspond to a road network in which there is exactly one path between two port towns that does not visit the same place twice. All roads can be used in both directions.

-----Output-----
Output one line with the average distances in kilometres between all pairs of distance signs in Optimistan. Your answer should have an absolute or relative error of at most $10^{-9}$.

If it is impossible to determine the exact average of distances between all pairs of distance signs in Optimistan, output “impossible”.

-----Examples-----
Sample Input:
3
4 4
2
Sample Output:
2.13333333333333
user: Simon Haples is a somewhat peculiar person. Not quite hip, not quite square, he is more of a triangular nature: ever since childhood, he has had an almost unhealthy obsession with triangles. Because of his discrete nature, Simon’s favorite kind of triangles are the Pythagorean ones, in which the side lengths are three positive integers $a$, $b$, and $c$ such that $a \le b$ and $a^2 + b^2 = c^2$. 

Recently, Simon has discovered the fantastic world of counting modulo some integer $n$. As you may imagine, he quickly realizes that there are multitudes of Pythagorean triples to which he has previously been oblivious! Simon therefore sets out to find all Pythagorean triples modulo $n$, i.e., all triples of integers $a$, $b$ and $c$ between $1$ and $n-1$ such that $a \le b$ and $a^2 + b^2 \equiv c^2 \pmod{n}$.

As Simon’s best friend, you realize that there is not much hope in deterring Simon from his crazy plans, so you decide to help him by computing how many such triples there are, so that Simon will know when his work is done.

-----Input-----
The input consists of a single integer $n$, satisfying $2 \le n \le 500000$.

-----Output-----
Output the number of Pythagorean triples modulo $n$.

-----Examples-----
Sample Input:
7
Sample Output:
18
user: Frank N. Stein is a very conservative high-school teacher. He wants to take some of his students on an excursion, but he is afraid that some of them might become couples. While you can never exclude this possibility, he has made some rules that he thinks indicates a low probability two persons will become a couple:
 - Their height differs by more than $40$ cm.
 - They are of the same sex.
 - Their preferred music style is different.
 - Their favourite sport is the same (they are likely to be fans of different teams and that would result in fighting).

So, for any two persons that he brings on the excursion, they must satisfy at least one of the requirements above. Help him find the maximum number of persons he can take, given their vital information.

-----Input-----
The first line of input consists of an integer $1 \le N \le 1000$ giving the number of pupils. Next there will be one line for each pupil consisting of four space-separated data items:
 - an integer $h$ ($0 \le h \le 3000$) giving the height in cm;
 - a character ‘F’ for female or ‘M’ for male;
 - a string describing the preferred music style;
 - a string with the name of the favourite sport.

The strings in the input are of length at at least $1$ and at most $20$ characters, and contain only upper and lower case letters, digits, and dashes.

-----Output-----
For each test case in the input there should be one line with an integer giving the maximum number of eligible pupils.

-----Examples-----
Sample Input:
4
35 M classicism programming
0 M baroque skiing
43 M baroque chess
30 F baroque soccer
Sample Output:
3
user: Mirko is playing with stacks. In the beginning of the game, he has an empty stack denoted with number $0$. In the $i$-th step of the game he will choose an existing stack denoted with $v$, copy it and do one of the following actions:
 - place number $i$ on top of the new stack
 - remove the number from the top of the new stack
 - choose another stack denoted with $w$ and count how many different numbers exist that are in the new stack and in the stack denoted with $w$

The newly created stack is denoted with $i$.

Mirko doesn’t like to work with stacks so he wants you to write a program that will do it for him. For each operation of type 2 output the number removed from stack and for each operation of type 3 count the required numbers and output how many of them there are.

-----Input-----
The first line of input contains the integer $N$ ($1 \leq N \leq 300000$), the number of steps in Mirko’s game.

The steps of the game are chronologically denoted with the first $N$ integers.

The $i$-th of the following $N$ lines contains the description of the $i$-th step of the game in one of the following three forms:
 - a v for an operation of type 1.
 - b v for an operation of type 2.
 - c v w for an operation of type 3.

The first character in the line denotes the type of operation and the following one or two denote the accompanying stack labels that will always be integers from the interval $[0, i - 1]$.

For each operation of type 2, the stack we are removing the element from will not be empty.

-----Output-----
For each operation type 2 or 3 output the required number, each in their own line, in the order the operations were given in the input.

-----Examples-----
Sample Input:
5
a 0
a 1
b 2
c 2 3
b 4
Sample Output:
2
1
2
user: In the rhythm game Arcaea, the player can play songs with various partners. Every partner has two skill levels: Frag, and Step. A larger Frag allows playing songs to earn a larger number of fragments, the main resource of the game. A larger Step helps the player progress faster in the World Mode that unlocks new songs and game stories. Some of the partners can optionally be awakened. After being awakened, a partner’s Frag and Step both reach a higher level.

A keen player of Arcaea is likely to have obtained many partners. But as one can only be accompanied by one partner when playing songs, partners with higher Frag and Step values are generally favored. We say a partner $x$ is more favorable than another partner $y$, if $x$’s Frag value is strictly larger than $y$’s Frag value, and $x$’s Step value is strictly larger than $y$’s Step value. Additionally, for a set $S$ of partners we define $d(S)$, the diversity of $S$, to be the maximum number of partners one can choose from $S$, so that no chosen partner is more favorable than another chosen partner.

Seine is an Arcaea player who has a set $S$ of $n$ unawakened partners. Seine wants to choose at most $k$ partners to awaken to maximize $d(S)$. What is the maximum $d(S)$ she can achieve?

-----Input-----
The first line of the input has two integers $n$ and $k$ ($1 \leq k \leq n \leq 2000$). The next $n$ lines each have four integers $g, p, g_ a, p_ a$ to describe one partner’s skill levels. The first two integers $g, p$ are the partner’s Frag and Step values before being awakened ($1 \leq g, p \leq 10^9$). The last two integers $g_ a, p_ a$ are zeroes if the partner cannot be awakened. Otherwise $g_ a, p_ a$ give the partner’s new Frag and Step values after being awakened, and it is guaranteed that $g < g_ a \leq 10^9$ and $p < p_ a \leq 10^9$.

-----Output-----
Output the maximum $d(S)$ Seine can achieve, if she can awaken up to $k$ partners. Note that Seine may choose to not awaken any partner.

-----Note-----
Arcaea is created and developed by Lowiro Limited. Lowiro does not endorse and has no involvement with the ProgNova contest.

-----Examples-----
Sample Input:
6 1
78 61 88 71
80 80 90 90
70 90 80 100
90 70 0 0
80 67 0 0
90 63 0 0
Sample Output:
5