meetforever/GRPO_CL huggingface dataset:
user: A friend of yours works at an undisclosed company in the music streaming industry, and needs your help. The company has offices in Stockholm and London, and collaboration between the two offices is extensive. The situation is that each of the many but small projects are handled by a two-person team with a member in each city. While emails, faxes, and phones are wonderful, and work well within each team, the CEO wants a briefing every year on the projects. For this purpose the CEO invites representatives from the projects to Barbados for a week of beach fun presentations of all the projects. 

However, money is tight and a new policy has been created: the CEO wants at least one person from each project, and additionally, she wants to invite as few people as possible. This is where you come in. In order to help your friend get a ticket to Barbados, you are to write a program that, given all the two-person teams, computes the smallest number of people that must be invited in order to get at least one person from each project, as well as a list of people to invite. If possible (subject to the set of people being smallest possible), the list of invitees should include your friend.

-----Input-----
The first line of input contains an integer $1 \le m \le 10000$, the number of teams. The following $m$ lines each contain two integers, $i,j$ separated by a space, being the employee IDs of the two employees in that team (the first one is from Stockholm and the second one is from London). Stockholm employees have IDs in the range $1000$ to $1999$ and London employees have IDs in the range $2000$ to $2999$. An employee can be a member of several teams, but there cannot be several teams consisting of the same pair of employees. Your friend has ID $1009$.

-----Output-----
Output first a single line with an integer $k$ indicating the smallest number of employees that must be invited to meet the requirements above. Then output $k$ lines giving the IDs of employees to invite. If possible (subject to $k$ being smallest possible), the list should contain your friend.

If there are several solutions subject to these constraints, anyone is acceptable.

-----Examples-----
Sample Input:
2
1009 2011
1017 2011
Sample Output:
1
2011
user: Fatima commutes from KTH to home by subway every day. Today Robert decided to surprise Fatima by baking cookies and bringing them to an intermediate station. Fatima does not always take the same route home, because she loves to admire the artwork inside different stations in Stockholm. However, she always optimizes her travel by taking the shortest route. Can you tell Robert which station he should go to in order to surely intercept Fatima?

-----Input-----
The first line contains two integers $N$ and $M$, $1 \leq N,M \leq 100000$, where $N$ is the number of subway stations and $M$ is the number of subway links. $M$ lines follow, each with three integers $u$, $v$, $w$, $0 \leq u,v < N$, $0 < w \leq 1000000000$, meaning that there is a one-way link from $u$ to $v$ that takes $w$ seconds to complete. Note that different subway lines may serve the same route.

The last line contains two integers $s$ and $t$, $0 \leq s,t < N$ the number of the station closest to KTH and closest to home, respectively. It is possible to reach $t$ from $s$.

-----Output-----
A space separated list of all the station numbers $u$ such that all shortest paths from $s$ to $t$ pass through $u$, in increasing order.

-----Examples-----
Sample Input:
4 4
0 1 100
0 2 100
1 3 100
2 3 100
0 3
Sample Output:
0 3
user: Every year the three friendly teachers at the kindergarten let their classes’ kids change classes to form three new ones. Some kids, of course, are old enough to leave, but those who stay for another year are rearranged among the three teachers.

The teachers even let the kids have their say in the process. As friendship both comes and goes hastily in young years, each kid $X$ ranks every other kid $Y$ according to how glad $X$ would be to have $Y$ in her new class. In fact, $X$ produces a preference list giving a total order of the other kids, i.e. there are no such things as ties – kids she would be equally glad to have as classmates.

The three teachers do not mind if the new classes formed are unbalanced in size, since they fill up their classes with new kids about to start their first year at the kindergarten. They do, however, want all the kids in their own new class to be different from last year, since even a teacher needs a break after a whole year with the same kids. They decide that a best partition into three classes under these premises is one where no kid is in the same class as a kid not listed among the top $T$ entries on their preference list, for $T$ as small as possible. Note that the kids in a new class may very well be the same as in an old one, but then with a new teacher!

-----Input-----
The first line of input contains an integer $1\leq N \leq 200$ giving the number of kids to be rearranged at the kindergarten. The kids are numbered $1$ through $N$.

Then follow $N$ lines describing the kids. The $i$-th row first contains the identifier of their current class’ teacher (an integer $0$, $1$, or $2$), and next the $N-1$ integers $\{ 1,2,3,…,i-1,i+1,…,N\} $ in some order, describing the classmate preference list of the $i$-th kid, in descending order.

-----Output-----
The smallest non-negative integer $T$, such that there is a partitioning of the kids in three new classes such that
 - no kid has the same teacher as in their current class, and
 - all kids’ classmates are among the top $T$ places of their preference lists, respectively.

-----Examples-----
Sample Input:
6
0 2 3 4 5 6
0 1 3 4 5 6
1 6 5 4 2 1
2 6 5 3 2 1
1 1 2 3 4 6
2 1 2 3 4 5
Sample Output:
4
user: A grasshopper is in a flower field. The field contains $N \cdot N$ flowers arranged in $N$ rows and $N$ columns. For each flower in the field, we know how many petals it has.

The grasshopper is initially on the flower in row $R$ and column $C$. Its goal is to visit as many flowers as possible while obeying these rules:
 - It can only jump into an adjacent row or column. If it jumps into the adjacent row, it must jump at least two columns, and if it jumps into the adjacent column, it must jump at least two rows. In other words, it can jump from flower $(r_1 , c_1)$ to flower $(r_2, c_2)$ if:
 - $|r_1 -r_2| = 1$ and $|c_1 -c_2 |> 1$ or
 - $|c_1 -c_2| = 1$ and $|r_1 -r_2 |> 1$
 - The number of petals on the next flower must be strictly larger than the number of petals on the previous flower.

Write a program that calculates the largest number of flowers the grasshopper can visit.

-----Input-----
The first line contains the integer $N$ ($1 \le N \le 1500$), the size of the field.

The second line contains integers $R$ ($1 \le R \le N$) and $C$ ($1 \le C \le N$), the grasshopper’s initial position.

The next $N$ lines contain $N$ positive integers separated by spaces, each at most $1000000$, the numbers of petals on the flowers.

-----Output-----
Output a single integer – the largest number of flowers the grasshopper can visit.

-----Examples-----
Sample Input:
4
1 1
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
Sample Output:
4
user: A cellular provider has installed $n$ towers to support their network. Each tower provides coverage in a $1\text { km}$ radius, and no two towers are closer than $1\text { km}$ to each other. The coverage region of this network is therefore the set of all points that are no more than $1\text { km}$ away from at least one tower. The provider wants as much of this region as possible to be connected, in the sense that a user at any point within a connected subregion can travel to any other point within the connected subregion without having to exit the subregion. Their current installation of towers may or may not already form a single connected region, but they have the resources to build one more tower wherever they want, including within $1\text { km}$ of an existing tower. Given that the provider is able to build one more tower, what is the maximum number of towers (including the new one) that can be included within a single connected subregion of coverage?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input consists of a single integer $n$ ($1 \le n \le 5000$) denoting the number of existing towers. Next follow $n$ lines, each with $2$ space-separated floating-point numbers $x$ and $y$ ($0 \le x, y \le 100000$, at most $5$ digits after the decimal point), denoting the location of a tower in km. It is guaranteed that the optimal number of towers will not change even if the coverage radius of all the towers is increased or decreased by $10^{-6}\text { km}$.

-----Output-----
Output a single integer, denoting the maximum number of towers that can be within a single connected subregion of the network after installing one additional tower.

-----Examples-----
Sample Input:
5
1.0 1.0
3.1 1.0
1.0 3.1
3.1 3.1
4.2 3.1
Sample Output:
6
user: A test for allergy is conducted over the course of several days, and consists of exposing you to different substances (so called allergens). The goal is to decide exactly which of the allergens you are allergic to. Each allergen has a live duration $D$ measured in whole days, indicating exactly how many days you will suffer from an allergic reaction if you are allergic to that particular substance. An allergic reaction starts to show almost immediately after you have been exposed to an allergen which you are allergic to. The test scheme has two action points per day:
 - At 8 o’clock each morning, at most one of the allergens is applied to your body.
 - At 8 o’clock each evening, you are examined for allergic reactions.

Thus an allergen with live duration $D$ will affect exactly $D$ allergic reaction examinations.

Of course, if you have two or more active allergens in your body at the time of an observed reaction, you cannot tell from that information only, which of the substances you are allergic to.

You want to find the shortest possible test scheme given the durations of the allergens you want to test. Furthermore, to allow simple large scale application the test scheme must be non-adaptive, i.e. the scheme should be fixed in advance. Thus you may not choose when to apply an allergen based on the outcome of previous allergic reaction examinations.

-----Input-----
The first line of the input contains a single integer $k$ ($1 \leq k \leq 20$) specifying the number of allergens being tested for. Then follow $k$ lines each containing an integer $D$ ($1 \leq D \leq 7$) specifying the live duration of each allergen.

-----Output-----
The number of days of the shortest conclusive non-adaptive test scheme.

A scheme ends the morning when you no longer have active allergens in your body, thus a test scheme for a single allergen with live duration $D$ takes $D$ days.

-----Examples-----
Sample Input:
3
2
2
2
Sample Output:
5
user: Note that this is a harder version of the problem pikemaneasy.Programming is an ancient art. Archeologists have made findings which indicate that already in the Middle Ages, infantry were practicing for programming contests while not in battle. Although it is not known how the programming contests were carried out (there were no computers), the archeologists have come up with a good theory (according to them). It states that infantry submitted pseudocode carved into stone, and then by the end of the contest, a genius priest named Kátisse ran all the programs in her head for correction. How they know her name? They won’t say. 

One of the reasons for this somewhat peculiar theory was the finding of ancient pike, a combat spear. Scientists have found many of these throughout the years. They come with a special symbol carved into them, usually the symbol of the tribe. This one didn’t have a symbol carved into it, it had pseudo code for Fenwick trees, as well as a config file for some kind of editor. Scientists are unsure which editor it might have been, but they believe it was some version of the closed Emacs beta.

Instead of looking for more evidence, the archeologists started speculating what strategy the pikemen used in these programming contests. They must have been well prepared, since this guy had algorithms carved into his spear. The contest rules were probably as follows: When submiting a solution to the judge, the time in minutes from contest start was added to a penalty counter. So in order to plan his problem solving, a pikeman must have been good at approximating the number of minutes required to solve each problem.

You are given a number of problems which were designed for a contest in which the pikeman participated. For each problem, you are given the estimated time in minutes for solving the problem. Calculate the maximum number of problems a pikeman can solve in the contest, and the minimum penalty he can get, under the assumptions that these estimations are correct. You may assume that the pikemen are very efficient: submissions are always correct, and after submitting a problem they start solving the next problem immediately.

-----Input-----
Input starts with two integers on a single line $1 \le N \le 10^9$ and $ 1 \le T \le 10^{18}$, the number of problems in the ancient contest and the total length of the contest in minutes. Then follows a line with four integers $1 \le A, B, C, t_0 \le 10^6$, where $t_0$$(t_0\leq C)$ specifies the time in minutes required for solving the first problem, and the rest of the times $t_1, \dots , t_{N-1}$ are given by:\begin{equation*} t_ i = ((At_{i-1}+B) \text {mod } C) + 1, i \in [1,N-1] \end{equation*}

-----Output-----
Output should consist of two integers: the maximum number of problems a pikeman can solve within the time limit, and the total penalty he will get for solving them. As the penalty might be huge, print it modulo $1000000007$. Print them on the same line, separated by a single space.

-----Examples-----
Sample Input:
1 3
2 2 2 1
Sample Output:
1 1
user: Having passed the Hunter Exam, Gon is now officially a Hunter! Gon is now saying goodbye to his best friend, Killua, as Gon wants to visit his home in ‘While Island’.

Gon and Killua plan to use the online chat application, Olaz, to keep in touch. However, they are not confident with Olaz’s security: some imposters may be able to login to Gon’s account and send messages to Killua! To prevent this, they have decided to use the following method:
 - Before starting a conversation, Gon must send Killua a number $X$ with exactly $n$ digits.
 - Killua must reply with an integer $Y$ with exactly $n$ digits, where $X$ and $Y$ form a best friend pair.
 - Each time they start a conversation, they must use a different best friend pair. This would avoid imposters from simply reusing the previous best friend pairs.

To define a best friend pair, first we define a friendly operation on a number $X$ as follow:
 - Select two adjacent digits of $X$.
 - Either add $1$ to both digits, or subtract $1$ from both digits.
 - It is forbidden to add $1$ to digit $9$, or subtract $1$ from digit $0$.
 - It is also forbidden to subtract $1$ from the first digit of $X$, if the first digit of $X$ is $1$.

Note that the last two conditions guarantee that the new number is valid and does not have leading zero. The new and old numbers will also have the same length.

Two numbers $X$ and $Y$ without leading zeros are called best friends, if we can obtain $Y$ from $X$, by applying a finite number of friendly operations. Note that a number $X$ is best friend with itself.

For example, $666$ and $875$ are best friends because we can apply the operations as follow:
 - $666 \rightarrow 776$
 - $776 \rightarrow 886$
 - $886 \rightarrow 875$

Now Gon is wondering how many conversation can they have, before running out of best friend pairs.

-----Input-----
The input contains exactly one integer $n$ $(1 \le n \le 10^5)$.

-----Output-----
The output contains exactly one integer — the number of pairs of best friend numbers with exactly $n$ digits, modulo $998244353$.

-----Examples-----
Sample Input:
1
Sample Output:
10
user: UpCoder is looking to assign their best employees to a team tasked with designing their new and improved website, and they’re looking to you to help them form the team. There are $n$ potential candidates. The CEO is employee number $0$, and the candidates are all assigned employee numbers ranging from $1$ through $n$. Each candidate is recommended by an employee with a smaller employee number. Each candidate can be described by three numbers (in addition to their employee number): their negotiated salary, their expected productivity, and the number of the employee who recommended them.

You would like to assign exactly $k$ candidates out of the $n$ total candidates to the team. The total value that you can get from these candidates is the sum of their productivities divided by the sum of their salaries. Note that you may only assign a candidate to the team if their recommender is also part of the team, or is the CEO. So, at least one candidate that you assign needs to have the CEO as a reference. The CEO handles the business aspect of the company, so s/he will not be counted as part of the $k$ candidates chosen for the team.

Find the maximum total value your team can provide given these constraints.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of the input will consist of two space separated integers $k$ and $n$ ($1 \le k \le n \le 2500$), where $k$ is the size of the team you must form, and $n$ is the total number of employee candidates. Each of the following $n$ lines will hold three space-separated integers describing an employee. Employee $1$ will be described first, then employee $2$, and so on. The three integers are $s$, $p$ and $r$, where $s$ ($1 \le s \le 10000$) is the employee’s salary, $p$ ($1 \le p \le 10000$) is the employee’s productivity, and $r$ ($0 \le r<i$) is the employee number of the employee who recommended this candidate (where $i$ is the employee number of this candidate).

-----Output-----
Output a single real number, which represents the maximum total value you can achieve forming a team of $k$ employees, subject to the constraints of the problem. Output this number to exactly three decimal places, rounded (standard $5 \uparrow / 4 \downarrow $ rounding).

-----Examples-----
Sample Input:
1 2
1000 1 0
1 1000 1
Sample Output:
0.001
user: You are given a simple graph with $N$ nodes and $M$ edges. The graph has the special property that any connected component of size $s$ contains no more than $s + 2$ edges. You are also given two integers $k$ and $P$. Find the number of $k$-colourings of the graph, modulo $P$.

Recall that a simple graph is an undirected graph with no self loops and no repeated edges. A $k$-colouring of a graph is a way to assign to each node of the graph exactly one of $k$ colours, such that if edge $(u, v)$ is present in the graph, then $u$ and $v$ receive different colors.

-----Input-----
The first line of input consists of four integers, $N, M, k$, and $P$ ($1 \leq N \leq 50000$, $0 \leq M \leq 1.5 N$, $1 \leq k \leq 10^9$, $1 \leq P \leq 2 \cdot 10^9$). The next $M$ lines of input each contains a pair of integers $A$ and $B$ ($1 \leq A \leq N$, $1 \leq B \leq N$), describing an edge in the graph connecting nodes $A$ and $B$.

-----Output-----
Output the number of $k$-colourings of the given graph, modulo $P$.

-----Examples-----
Sample Input:
3 3 2 10000
1 2
2 3
3 1
Sample Output:
0
user: You found a complete, undirected graph with $n$ nodes, labeled $1$ to $n$. Each edge has a color. For simplicity, each color is identified by a number between $1$ and $300$ inclusive. Interestingly, you noticed that for each and every simple cycle in this graph, there are at least two adjacent edges on this cycle which have the same color.

For each non-empty subset of nodes in graph $S$, let $f(S)$ denote the size of the maximum subset of nodes you can choose from $S$ such that all edges between the chosen nodes are the same color. Compute the sum of $f(S)$ over all non empty subsets $S$ of nodes in the graph.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input will contain a single integer $n$ ($1 \le n \le 300$), which is the number of nodes in the graph.

The next $n$ lines will each contain $n$ integers $c$ ($0 \le c \le 300$), which is a matrix representing the colors of the edges, where $c[x,y]$ is the color of the edge between node $x$ and node $y$. It is guaranteed that the values on the diagonal will be $0$ ($c[x,x]=0$), since there is no edge from a node to itself. It is also guaranteed that the matrix is symmetric and the off-diagonal colors range from $1$ to $300$ ($1 \le c[x,y] = c[y,x] \le 300$ for $x \ne y$).

-----Output-----
Output a single integer, which is the sum of $f(S)$ over all non empty subsets $S$ of nodes in the graph. Since this number may be very large, output it modulo $10^9+7$.

-----Examples-----
Sample Input:
4
0 1 1 1
1 0 2 2
1 2 0 3
1 2 3 0
Sample Output:
26
user: You have been hired by Addictive Coin Machines to help design the next hit in their line of eye-catching, coin-guzzling, just-one-more-try Pachinko machines for casinos around the world.

Playing a Pachinko machine involves launching balls into a rectangular grid filled with pegs, obstacles, and targets. The ball bounces around the grid until it eventually hits one of the targets. The player earns a certain number of points depending on which target is hit.

The grid pattern for the next Pachinko machine has already been designed, but point values for the targets have not been assigned. These must be set so that like all casino machines, the machine is profitable but not too profitable. Thus it is important to figure out the probability of a ball hitting any particular target. That’s your job!

For simplicity, the grid is modeled as a tall rectangle filled with mostly-open spaces (each represented by ‘.’), impassable obstacles (each represented by ‘X’), and targets (each represented by ‘T’).

A ball is launched randomly with uniform probability into one of the mostly-open spaces on the top row of the grid. From that point on, collisions with pegs cause the ball to randomly bounce up, down, left, or right, with various given probabilities. For simplicity, assume these probabilities are the same for every space in the grid. If the ball bounces into an obstacle or attempts to move off the grid, it won’t actually move from its current space. When the ball moves into a target it is removed from play.

You can safely assume that the average number of spaces visited by a ball before hitting a target will not exceed $10^{9}$. It would not make for a very enjoyable game if the ball just bounces forever!

For each target, calculate the probability that it is the one hit by a launched ball.

-----Input-----
The input consists of a single test case. The first line contains integers $w$ and $h$, which are the width and height of the Pachinko grid ($1 \leq w \leq 20$ and $2 \leq h \leq 10000$). The next line contains four non-negative integers $u$, $d$, $l$, and $r$, which sum to 100 and are the percentage probabilities of the ball bouncing up, down, left, or right from any open space.

Each of the next $h$ lines contains $w$ characters, each of which is ‘.’, ‘X’, or ‘T’. These lines describe the Pachinko grid. The first line, which describes the top row of the grid, contains at least one ‘.’ and no ‘T’s.

-----Output-----
Display one line for each ‘T’ in the grid, in order from top to bottom, breaking ties left to right. For each target, display the probability that a launched ball will hit it. Give the answer with an absolute error of at most $10^{-6}$.

-----Examples-----
Sample Input:
3 2
20 20 20 40
X.X
T.T
Sample Output:
0.333333333
0.666666667