meetforever/GRPO_CL huggingface dataset:
user: The $n$-queens problem is popular among computer science students. Its task is to place a maximum number of queens on an $n\times n$ chessboard so that no two queens attack each other. Two queens attack each other if they share a row, column, or diagonal.

One day Zapray gets bored and extends the problem by loosening its constraint: It is allowed to have two queens attack each other, but no three queens can attack each other at the same time. In other words, there should not exist three queens $i$, $j$, and $k$ so that $i$ attacks $j$, $j$ attacks $k$, and $k$ attacks $i$.

With the new constraint, Zapray would like to know at most how many queens he can place on an $n \times m$ chessboard, and in how many ways he can place a maximum number of queens. Additionally, some of the cells on the chessboard are broken, and queens cannot be placed on these broken cells. A broken cell between two queens does not prevent them from attacking each other.

-----Input-----
The first line has two integers $n$ and $m$ ($1 \leq n, m \leq 50, n \cdot m \leq 50$), describing the size of the chessboard. Each of the next $n$ lines has $m$ characters describing the cells on one row of the chess board. A dot (‘.’) denotes an empty cell, while a hash (‘#’) denotes a broken cell. There is at least one usable cell on the chessboard.

-----Output-----
Output two integers. The first is the maximum number of queens Zapray can place on the chessboard, and the second is the number of ways in which he can place that many queens.

-----Examples-----
Sample Input:
3 4
....
.#.#
....
Sample Output:
5 13
user: In the year 2120 there is a vast subway network under all of Lund, consisting of $N$ stations and $M$ tunnels. Each tunnel connects two stations and the stations are numbered $1$, $\ldots $, $N$.

Erik has had enough of Skånetrafiken’s terrible route planning software and plans to build his own. To do this, he needs to know the length of each of the tunnels, but the subway map is incomplete in this regard. By looking out the window, Erik has noticed that some tunnels have special cables running alongside them, probably for providing power to the stations. The cables connect the stations so that every station is connected to the central station (the station numbered $1$). Knowing how greedy Skånetrafiken is, he is certain that the cables are placed so that the total length of cable is minimized.

Erik knows the precise length of some tunnels, and which tunnels contain cables. Using this information he wants to find the minimum possible length for each tunnel with unknown length. Unfortunately, Erik’s algorithm isn’t efficient enough to process the enormous size of Lund’s subway network. Can you help him by implementing a more efficient algorithm?

-----Input-----
The first line of input contains two integers $N$ and $M$, where $2 \leq N \leq 10^5$ and $N - 1 \leq M \leq 2 \cdot 10^5$, the number of stations and the number of tunnels, respectively. Each of the next $M$ lines contains the values $a_ i$, $b_ i$, $l_ i$ and $c_ i$. The integers $a_ i$ and $b_ i$, with $1 \leq a_ i, b_ i \leq N$ and $a_ i\neq b_ i$, denote the two stations connected by the $i$th tunnel. The value $l_ i$ is either an integer satisfying $1 \leq l_ i \leq 10^9$, the length of the $i$th tunnel if it is known, or a question mark “?”. Finally, $c_ i$ is $1$ if the $i$th tunnel contains a cable, and $0$ if not.

It is guaranteed that there is at most one tunnel connecting the same pair of stations, and that it is possible to travel between any pair of stations using the subway. It is also guaranteed that there exists a path between any station and station number $1$ using only tunnels where $c_ i = 1$.

-----Output-----
For each tunnel with $l_ i=\texttt{?}$, output one line with a single integer, the minimum possible length for that tunnel. Tunnel lengths should be output in the same order as the tunnels are listed in the input.

-----Sample Description-----
In the first sample case, the minimal distance for the unknown tunnel (between stations $3$ and $1$) is $5$. This is because, if the length were less than $5$, it would be more efficient for Skånetrafiken to run cables through the second and third tunnels.

-----Examples-----
Sample Input:
3 3
1 2 5 1
2 3 3 1
3 1 ? 0
Sample Output:
5
user: You own and manage an office building. One of the tenants, a regional paper supply company, is complaining about the lack of security offered by the CrapTacular$^{\text {TM}}$ doors you installed in the building. A potential attacker could easily pick the locks on these doors to gain access to any room of the building. This is frankly unacceptable as there are reams upon reams of priceless paper stored inside the office.

To appease your tenant, you offer to replace one existing door with a new high-security door that will stop any attacker. To get the most benefit from the new door, it should protect the maximum possible number of rooms in the building. That is, you need to find which existing door to replace with a secure door such that the maximum number of rooms can only be reached from outside the building through the new door.

The building contains $0<N<100000$ rooms which are connected by $0<M<100000$ doors. Rooms are uniquely named by the integers in the range $[0,N)$. To figure out where to install the secure door, you have a floor plan of the building. The floor plan somehow represents all of the floors in one (possibly non-planar) graph. The only way to pass from one room to another, or the outside, is by using a door. All rooms can be reached, directly or via other rooms, from all other rooms and from the outside of the building.

-----Input-----
Input contains several lines of integers separated by spaces. The first line contains the number $N$, then $M$. The following $M$ lines describe each door with two numbers $-1 \le u < N$ and $-1 \le v < N$, $u \ne v$. The numbers $u$ and $v$ represent the rooms connected by the door. A door connects to the outside if $u$ or $v$ is $-1$. No door has $u = v$.

-----Output-----
The maximum number of rooms that can be protected with a single high-security door.

-----Sample Explanation-----
The image below illustrates the building described in Sample Input 2:





Here, double-headed arrows indicate doors. For example, the arrow between room $0$ and $1$ shows that there is a door there. The other arrow connecting room $0$ leads to the outside of the building. We could totally secure room $5$ by replacing the door between room $1$ and $5$, but this is not the optimal solution. The best solution for this input is to replace the door between room $1$ and $2$, since room $2$, $3$, and $4$ can only be reached from the outside of the building by passing through that door. The answer is thus $3$ because at most $3$ rooms can be secured.

-----Examples-----
Sample Input:
2 3
-1 0
-1 1
0 1
Sample Output:
0
user: In 1978 AD the great Sir Isaac Newton, whilst proving that $\mathcal{P}$ is a strict superset of $\mathcal{NP}$, defined the Beta Alpha Pi Zeta function $f$ as follows over any sequence of positive integers $a_1, \dots , a_ n$. Given integers $1\leq i\leq j\leq n$, we define $f(i, j)$ as $\gcd (a_ i, a_{i+1}, \dots , a_{j-1}, a_ j)$. 

About a century later Lothar Collatz applied this function to the sequence $1, 1, 1, \dots , 1$, and observed that $f$ always equalled $1$. Based on this, he conjectured that $f$ is always a constant function, no matter what the sequence $a_ i$ is. This conjecture, now widely known as the Collatz Conjecture, is one of the major open problems in botanical studies. (The Strong Collatz Conjecture claims that however many values $f$ takes on, the real part is always $\frac{1}{2}$.)

You, a budding young cultural anthropologist, have decided to disprove this conjecture. Given a sequence $a_ i$, calculate how many different values $f$ takes on.

-----Input-----
The input consists of two lines.
 - A single integer $1 \leq n \leq 5 \cdot 10^5$, the length of the sequence.
 - The sequence of integers $a_1, a_2, \dots , a_ n$. It is given that $1 \leq a_ i \leq 10^{18}$.

-----Output-----
Output a single line containing a single integer, the number of distinct values $f$ takes on over the given sequence.

-----Examples-----
Sample Input:
4
9 6 2 4
Sample Output:
6
user: Recently, Anthony has really gotten into the battle royale games. His favorite game is PUBNite.

In PUBNite, many players fight against each other and try to be the last one alive. One of the most well-known mechanics in PUBNite is the safety zone, which is a circular area. Players outside of the safety zone will take one damage per second continuously.

In PUBNite the safety zone is shrinking at the start of the game. The safety zone is centered at coordinate $(x_ s, y_ s)$ and has an initial radius of $r_ i$ meters. The radius is decreasing at a constant and continuous rate of $s_ s$ m/s (meters per second) and will stop decreasing once it’s no more than $r_ f$ meters. Anthony is initially located at coordinate $(x_ a, y_ a)$ and can run at a maximum speed of $s_ a$ m/s. What is the minimum amount of damage that Anthony will take assuming the safety zone never shrinks again after its radius is no more than $r_ f$?

-----Input-----
The first line of the input contains integers $x_ s, y_ s, s_ s, r_ i, r_ f$. You are guaranteed that $|x_ s|, |y_ s|\leq 10^9$, $0<s_ s\leq 10^9$, and $0<r_ f\leq r_ i\leq 10^9$.

The second line of the input contains integers $x_ a, y_ a, s_ a$. You are guaranteed that $|x_ a|, |y_ a|\leq 10^9$ and $0<s_ a\leq 10^9$.

-----Output-----
A single number $d$ denoting the minimum amount of damage Anthony will take. You are required to output $d$ with absolute precision $10^{-6}$.

-----Examples-----
Sample Input:
2 1 1 3 2
2 5 1
Sample Output:
2.000000
user: A network of size $N$ contains $N$ computers connected by $N-1$ cables, so that there is exactly $1$ path between any pair of computers.

The transmission cost between $2$ computers is equal to the square of the number of cables on the path connecting the $2$ computers.

The transmission cost of a network is equal to the sum of the transmission cost between all unordered pair of computers.

Given network $A$ with $N$ computers and network $B$ with $M$ computers, the administrator wants to create a new network $C$, by adding exactly one cable connecting one computer in $A$ and one computer in $B$.

Your task is to minimize the transmission cost of the new network $C$.

-----Input-----
 - The first line contains an integer $N$ - the number of computers in the network $A$ ($1 \leq N \leq 50000$).
 - In the next $N-1$ lines, each line contains two distinct integers $u$ and $v$, representing a cable connecting computers $u$ and $v$ in network $A$ ($1 \leq u,v \leq N$).
 - The next line contains an integer $M$ - the number of computers in the network $B$ ($1 \leq M \leq 50000$).
 - In the next $M-1$ lines, each line contains two distinct integers $u$ and $v$, representing a cable connecting computers $u$ and $v$ in network $B$ ($1 \leq u,v \leq M$).

It is guaranteed that each network is a tree.

-----Output-----
Write in one line the minimum transmission cost of the resulting network $C$.

-----Sample clarification-----
In the first sample below, connecting computer $2$ of network $A$ and computer $1$ of network $B$ will minimize the transmission cost of the network.

In the second sample below, connecting computer $4$ of network $A$ and computer $1$ of network $B$ will minimize the transmission cost of the network.

-----Examples-----
Sample Input:
3
1 2
2 3
4
1 2
1 3
1 4
Sample Output:
96
user: This is a very simple problem. You are given $N$ points. Some points may be repeated. The weight (distance) between two points is given by the Manhattan distance between the two points. Find the weight of a Minimum Spanning Tree that spans these $N$ points.

-----Input-----
The input consists of:
 - One line with one integer $N$ ($1 \leq N \leq 100000$), the number of points,
 - $N$ lines each with two integers $x$ and $y$ ($0 \leq x,y < 1000$), the coordinates of each point.

-----Output-----
Output one line with a single integer: The weight of a Minimum Spanning Tree that spans these $N$ points.

-----Examples-----
Sample Input:
4
0 0
0 1
1 0
1 1
Sample Output:
3
user: Mirko and Slavko are playing a new game. Again. Slavko starts each round by giving Mirko two numbers $A$ and $B$, both smaller than 100. Mirko then has to slove the following task for Slavko: how to pair all given $A$ numbers with all given $B$ numbers so that the maximal sum of such pairs is as small as possible.

In other words, if during previous rounds Slavko gave numbers $a_1, a_2 ,a_3, \ldots , a_ n$ and $b_1, b_2, b_3, \ldots , b_ n$ , determine $n$ pairings $(a_ i, b_ j)$ such that each number in the $A$ sequence is used in exactly one pairing, each number in the $B$ sequence is used in exactly one pairing, and the maximum of all sums $a_ i + b_ j$ is minimal.

-----Input-----
The first line of input contains a single integer $N$ ($1 \leq N \leq 100000$), the number of rounds.

The next $N$ lines contain two integers $A$ and $B$ ($1 \leq A, B \leq 100$), the numbers given by Slavko in that round.

-----Output-----
The output consists of $N$ lines, one for each round. Each line should contain the smallest maximal sum for that round.

-----Examples-----
Sample Input:
3
2 8
3 1
1 4
Sample Output:
10
10
9
user: Joe works in a maze. Unfortunately, portions of the maze have caught on fire, and the owner of the maze neglected to create a fire escape plan. Help Joe escape the maze.

Given Joe’s location in the maze and which squares of the maze are on fire, you must determine whether Joe can exit the maze before the fire reaches him, and how fast he can do it.

Joe and the fire each move one square per minute, vertically or horizontally (not diagonally). The fire spreads all four directions from each square that is on fire. Joe may exit the maze from any square that borders the edge of the maze. Neither Joe nor the fire may enter a square that is occupied by a wall.

-----Input-----
The first line of input contains the two integers $R$ and $C$, separated by spaces, with $1 \le R, C \le 1000$. The following $R$ lines of input each contain one row of the maze. Each of these lines contains exactly $C$ characters, and each of these characters is one of:
 - #, a wall
 - ., a passable square
 - J, Joe’s initial position in the maze, which is a passable square
 - F, a square that is on fire

There will be exactly one J in the input.

-----Output-----
Output a single line containing “IMPOSSIBLE” if Joe cannot exit the maze before the fire reaches him, or an integer giving the earliest time Joe can safely exit the maze, in minutes.

-----Examples-----
Sample Input:
4 4
####
#JF#
#..#
#..#
Sample Output:
3
user: In a two player card game, you have $n$ minions on the board and the opponent has $m$ minions. Each minion has a health between $1$ and $6$. 

You are contemplating your next move. You want to play an “Explosion” spell which deals $d$ units of damage randomly distributed across all minions. The damage is dealt one unit at a time to some remaining minion on the board. Each living minion (including your own) has the same chance of receiving each unit of damage. When a minion receives a unit of damage, its health is decreased by one. As soon as the health of a minion reaches zero, it is immediately removed from the board, before the next damage is dealt. If there are no minions left on the board, any excess damage caused by the spell is ignored.

Given the current health of all minions, what is the probability that the Explosion will remove all of the opponent’s minions? Note that it does not matter if all your own minions die in the process as well, and the damage continues to be dealt even if all your own minions are gone.

-----Input-----
The first line of input contains the three integers $n$, $m$, and $d$ ($1 \le n, m \le 5$, $1 \le d \le 100$). Then follows a line containing $n$ integers, the current health of all your minions. Finally, the third line contains $m$ integers, the current health of all the opponent’s minions. All healths are between $1$ and $6$ (inclusive).

-----Output-----
Output the probability that the Explosion removes all the opponent’s minions, accurate up to an absolute error of $10^{-6}$.

-----Examples-----
Sample Input:
1 2 2
2
1 1
Sample Output:
0.3333333333
user: In a planet far far away, an intelligence agency plans to send some spies to the Earth to investigate the life there. In order to ensure the secret and secure of this scouting campaign, the agency gives a secret ID to each member.

Each ID is a binary string of length $n$. To prevent enemies from infiltrating the agency, they choose a pattern $P$, which is a string of 1 and *, and decides that an ID is valid iff it satisfies this pattern $P$.

A binary string $S = s_1 s_2 \ldots s_ n$ satisfies a pattern $P = p_1 p_2 \ldots p_ m$ iff any of these conditions holds:
 - $m = n$, and for each valid index $i$, either $s_ i = 1$ or $p_ i = *$.
 - $m < n$, and at least one substring of $S$ satisfies the pattern $P$.

For example:
 - Both strings 101 and 111 satisfy the pattern 1*1.
 - These strings 0101110, 1110000 and 1010111 satisfy the pattern 1*1, since 101 and 111 are their substrings.
 - The string 0100010 does not satisfy the pattern 1*1.

The agency wants to know how many spies they can employ, if each member has a unique ID.

-----Input-----
 - The first line contains one integer $n$ $(1 \leq n \leq 50)$ — the length of a valid ID.
 - The second line contains a string of at most $30$ characters 1 and *, where at least half of them are 1s — the pattern $P$ which all valid IDs must satisfy.

-----Output-----
Print a single integer — the maximum number of members in the agency, if each is assigned a unique valid ID.

-----Examples-----
Sample Input:
10
1
Sample Output:
1023
user: We define an infinite two dimensional array $F$ using the following formula:
 - $F_{0, 0} = 0$,
 - $F_{0, 1} = F_{1, 0} = 1$,
 - For $i \ge 2$, $F_{i, 0} = F_{i-1, 0} + F_{i-2, 0}$,
 - For $i \ge 2$, $F_{0, i} = F_{0, i-1} + F_{0, i-2}$,
 - For $i, j \ge 1$, $F_{i, j} = F_{i-1, j} + F_{i, j-1}$.

Here are the first few values of $F$:

Given $x$ and $y$, your task is to calculate $F_{x, y}$.

-----Input-----
The input consists of only a single line, containing $2$ integers $x$ and $y$ $(0 < x, y < 10^6)$.

-----Output-----
Print a single integer — the value of $F_{x, y}$, modulo $10^9 + 7$.

-----Examples-----
Sample Input:
2 2
Sample Output:
6
user: In the X-ray lab at KTH some experiments require evacuated tubes between source and sample and between sample and detector so that the X-rays are not absorbed by the air. Since the positions of object and detector vary between different experiments, several tubes of different lengths are available. The tubes should be fixed together in pairs, since they have a vacuum window only in one end. Two such tube pairs should be chosen, one to place between the source and the object and one to place between the object and the detector. This, however, gives a large set of possible lengths and makes it difficult to figure out which tubes to use for an experiment. The length of the tubes used should be as long as possible to minimize air absorption, but there is a limited amount of space between source and object $L_1$ and between object and detector $L_2$. What is the maximum length of air that can be replaced by vacuum tubes in this way? 

-----Task-----
Given a set of tube lengths and the two distances $L_1$ and $L_2$, find four tubes with the total length being as long as possible under the constraint that the sum of the first two tube lengths is at most $L_1$ and the sum of the last two tube lengths is at most $L_2$.

-----Input-----
The first line of input contains three positive integers, $L_1$ and $L_2$, denoting the distances explained above in mm ($1 \leq L_1, L_2 \leq 10000$) and $N$, the number of available tubes ($4 \leq N \leq 2000$). The following $N$ lines each contain a single positive integer less than or equal to $10000$, the length of a tube in mm.

-----Output-----
Output one line containing the maximum total length of air that can be avoided, i.e., the sum of the four tubes chosen. If there are no two pairs of tubes fitting into the setup, output the single word “Impossible” instead.

-----Examples-----
Sample Input:
1000 2000 7
100
480
500
550
1000
1400
1500
Sample Output:
2930
user: In a distributed system, data is never where you need it, and fetching data over a network takes time and consumes bandwidth. The problem can be mitigated by adding a cache, where a node stores some resources locally and if those resources need to be used again, it can simply take them from its cache rather than asking someone else for them. 

However, caches have a nasty tendency to fill up, so at some point, objects must be evicted from the cache to make room for new objects. Choosing what object to remove from the cache is not easy and there are several different algorithms to choose from.

The marvelous Apes in Computing Machinery have come up with a fantastic new algorithm, the Introspective Caching Algorithm, named after a city of Peru. It consists of some extra hardware (a very small, precognitive monkey) which helps making decisions. Since the monkey can see into the future, she knows exactly what objects will be accessed and in what order, and using this information she will make optimal decisions on what objects to remove from the cache. Optimality here means that she will minimize the number of times an object is read into the cache.

All object accesses go through the cache, so every time an object is accessed, it must be inserted into the cache if it was not already there. All objects are of equal size, and no writes occur in the system, so a cached object is always valid. When the system starts, the cache is empty.

You have been tasked with evaluating the monkey’s performance, and feeding her the occasional banana.

-----Input-----
The first line of input contains three integers, separated by single spaces, telling you how many objects fit in the cache, $0 < c \le 10000$, how many different objects are in the system, $c \le n \le 100000$, and how many accesses, $0 \le a \le 100000$, will occur. The following $a$ lines contain a single integer between $0$ and $n-1$ (inclusive) indicating what object is accessed. The first line corresponds to the first object accessed access and the last line to the last.

-----Output-----
Output the least number of times an object must be read into the cache to handle the accesses listed in the input.

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