meetforever/GRPO_CL huggingface dataset:
user: Arnar is playing his favourite video game Association of Myths. He’s playing a character by the name of Lumen. Lumen has a special ability where she can shoot a laser beam across the entire map killing all enemies it hits. The only drawback is that the beam is rather narrow and Arnar isn’t always sure where the enemies are located. He now needs your help to determine where the enemies are hiding so he can hit them (and hopefully win the game).

Arnar is very serious about this game and is therefore willing to meticulously research how the opponents move. He knows they only travel in a straight path called the ’medial path’. We can therefore describe their location as a function $f$ of a single variable. After much thought Arnar has concluded that $f$ can be described by\[ c\int _ a^b \left(t_1\Gamma (x) + \sqrt [t_2]{\log (\operatorname {erf}(t_3 x))} - J_ k(x)^{t_4}\right)dx \]

where $\log $ is the natural logarithm,\[ \Gamma (z) = \int _0^{\infty } x^{z - 1} e^{-x} dx, \]\[ \operatorname {erf}(x) = \frac{2}{\sqrt {\pi }} \int _0^x e^{-t^2} dt, \]

and\[ J_ k(x) = \frac{1}{\pi } \int _0^{\pi } \cos (k \tau - x \sin \tau ) d \tau . \]

Arnar thinks that it maybe a bit tough for you to compute $f$ as previously described so he tells you it is enough to calculate the $r$-th degree Taylor polynomial around $0$, i.e.\[ P(x) = \sum _{i = 0}^r \frac{f^{(i)}(0)}{i!}x^i. \]

Arnar is a afraid he was too aggressive when approximating $f$ with $P$ so he would like to modify $P$ a little bit further. He knows that as the game progresses his opponent will have more in-game currency and will therefore buy better, more agile shoes. To account for this Arnar recursively defines a sequence of polynomials by\[ P_0(x) = P(x), \quad P_ n(x) = \sum _{i = 0}^{r + n} P_{n - 1}(i) x^i. \]

Arnar finally notes that during the endgame phase the opponent will probably have to go to the bathroom and to account for that he wants reduce the size of the final polynomial in his sequence. He therefore takes $P_ s$ and differentiates it $\operatorname {deg}(P_ s) + 1$ times and calls the outcome $g$. Arnar is now satisfied that\[ \frac{(g(n) + l)^2}{\pi e} + \frac{1}{l + 1} \]

gives the location of his opponent. Why does Arnar take the last step? Well, Arnar won’t tell you because he’s afraid you will be able to beat him in Association of Myths if he tells you all his secrets. He also requests that you give him the answer to at least two correct decimal places since his mouse can track changes as small as a hundredth of an in-game unit of length.

-----Input-----
The first line of the input starts with three real numbers $a, b, c$, $-10^9 \leq a \leq b \leq 10^9$ and $1 \leq c \leq 10^9$. These numbers will have at most 6 digits after the decimal point. The second line has four integers $t_1, t_2, t_3, t_4$, $1 \leq t_ i \leq 10^9$. The third line has five integers $n, k, r, s, l$, $1 \leq n, k, r, s, l \leq 10^3$.

-----Output-----
The output should consist of one line containing the location of Arnar’s opponent as described above.

-----Examples-----
Sample Input:
-99.99 99.99 9999.99
99 9 999 9999
9 99 9 99 9
Sample Output:
9.585073
user: Instead of worrying about the current hyperinflation you decide to go down to the local bar and have a drink.

The prices at the bar are displayed using magnetic signs with numbers printed on them, with each magnet showing one or more digits. For instance, the price of $1106$ megacredits is displayed like this:


While the bartender is busy serving the next customer, you have just enough time to rearrange the price of your favourite beverage to make it as cheap as possible. But be quick about it!

Invidual magnets can be moved around in any order and turned upside-down. The numbers are shown in a script that makes it difficult for the bartender to distinguish $0$, $1$, and $8$ from their upside-down counterpart. Moreover, $6$ and $9$ look the same when one is turned upside-down. The example price above could be made almost ten times cheaper by turning the first magnet:


You have to use all the magnets, otherwise the bartender will immediately suspect foul play.

-----Input-----
On the first line, the number $n$ of magnets, with $1\in \{ 1,\ldots , 1000\} $. On each of the following $n$ lines, exactly one sequence $m_ i$ of digits describing the $i$th magnet. Each magnet $m_ i$ for $i\in \{ 1,\ldots , n\} $ consists of at least one and at most $10$ digits from $0$, $1$, $\ldots $, $9$. The price currently displayed on the bar is the integer described by the juxtaposition $m_1\cdots m_ n$ of the magnets in the order they are given, from left to right. Note that a magnet can be all $0$s, even though the current price at the bar, alas!, is certainly not.

-----Output-----
A single line containing the cheapest price formed by the magnets $m_1,\ldots ,m_ n$, rearranged in any order, and each of them possibly turned upside-down.

-----Examples-----
Sample Input:
2
110
6
Sample Output:
0116
user: “The Drinking Musicians”, a widely known and popular folk group, are coming to your town. The musicians are known not only by their playing skills, but also their rough character. They never arrive on time, don’t know which town they’re in, and frequently have trouble finding the stage.

Additionally, during the concert, each of the musicians at one point takes a break. If three or more of them are on a break at the same time, they start stirring trouble in town and the rest of the group start panicking and playing the wrong chords.

The concert will be $T$ minutes long, during which each of the $N$ members will take a break. The length of the break is known for each member.

Help the organizer of the concert by writing a program that determines how to schedule the breaks of the members so that, at any given moment, at most two are absent from the stage. All breaks must be entirely during the concert.

-----Input-----
The first line of input contains the integers $T$ and $N$ ($1 \le T \le 5000$, $1 \le N \le 500$), the length of the concert in minutes and the number of musicians in the group.

The next line contains $N$ integers (each between $1$ and $T$ inclusive) separated by single spaces, the length of the break in minutes for each member.

Note: The input data will be such that a solution, although not necessarily unique, will always exist.

-----Output-----
For each musician output one integer, the number of minutes the musician will spend on stage before going on the break. Output the musicians in the same order they were given in the input.

-----Examples-----
Sample Input:
8 3
4 4 4
Sample Output:
0 2 4
user: Basin City is known for her incredibly high crime rates. The police see no option but to tighten security. They want to install traffic drones at different intersections to observe who’s running on a red light. If a car runs a red light, the drone will chase and stop the car to give the driver an appropriate ticket. The drones are quite stupid, however, and a drone will stop before it comes to the next intersection as it might otherwise lose its way home, its home being the traffic light to which it is assigned. The drones are not able to detect the presence of other drones, so the police’s R&D department found out that if a drone was placed at some intersection, then it was best not to put any drones at any of the neighbouring intersections. As is usual in many cities, there are no intersections in Basin City with more than four other neighbouring intersections.

The drones are government funded, so the police force would like to buy as many drones as they are allowed to. Being the programmer-go-to for the Basin City Police Department, they ask you to decide, for a given number of drones, whether it is feasible to position exactly this number of drones.

-----Input-----
The first line contains an integer $k$ ($0 \leq k \leq 15$), giving the number of drones to position. Then follows one line with $1 \leq n \leq 100\; 000$, the total number of intersections in Basin City. Finally follow $n$ lines describing consecutive intersections. The $i$-th line describes the $i$-th intersection in the following format: The line starts with one integer $d$ ($0 \leq d \leq 4$) describing the number of intersections neighbouring the $i$-th one. Then follow $d$ integers denoting the indices of these neighbouring intersections. They will be all distinct and different from $i$. The intersections are numbered from $1$ to $n$.

-----Output-----
If it is possible to position $k$ drones such that no two neighbouring intersections have been assigned a drone, output a single line containing possible. Otherwise, output a single line containing impossible.

-----Examples-----
Sample Input:
4
7
2 2 4
3 1 3 5
1 2
2 1 5
4 2 6 4 7
2 5 7
2 6 5
Sample Output:
impossible
user: Misha needs to send packages to his friend Nadia. Both of them often travel across Russia, which is very large. So they decide to hire a messenger. Since the cost of the messenger service depends on the time it takes to deliver the package, they need your help to optimize a little bit.

Assume Misha and Nadia move on a two-dimensional plane, each visiting a sequence of places and moving along straight line segments from place to place. Your task is to find the shortest possible delivery time given their two paths.

Misha hands the package to the messenger at some point along his path. The messenger moves without delay along a straight line from the pick-up to intercept Nadia, who is traveling along her path. Misha, Nadia and the messenger move with a constant speed of $1$ distance unit per time unit. The delivery time is the time between Misha handing over the package and Nadia receiving it.

-----Input-----
The input consists of a single test case. The test case contains two path descriptions, the first for Misha and the second for Nadia. Each path description starts with a line containing an integer $n$, the number of places visited ($2 \leq n \leq 50000$). This is followed by $n$ lines, each with two integers $x_ i$ and $y_ i$ specifying the coordinates of a place ($0 \leq x_ i, y_ i \leq 30000$). Coordinates of the places are listed in the order in which they are to be visited, and successive places do not have the same coordinates.

Misha and Nadia start their journeys at the same time, visiting the places along their paths without stopping. The length of each path is at most $10^6$. The package must be picked up at the latest when Misha reaches his final place and it must be delivered at the latest when Nadia reaches her final place.

-----Output-----
Display the minimal time needed for delivery. Give the answer with an absolute error of at most $10^{-3}$ or a relative error of at most $10^{-5}$. If the package cannot be delivered, display impossible instead.

-----Examples-----
Sample Input:
2
0 0
0 10
2
4 10
4 0
Sample Output:
4.00000
user: Consider a square grid with lamps in fixed locations. Each lamp can either illuminate its row or its column, but not both. The illumination of each lamp extends over a limited distance.

Any square in the grid should only be illuminated by at most one lamp in its row and by at most one lamp in its column (one of each is acceptable, as is just the row, just the column, or neither). Determine if it is possible for all lamps to be lit while satisfying 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 input contains three positive integers, $n$, $r$ and $k$ ($1 \le n, r, k \le 1000, k \le n^2$), where $n$ is the size of one side of the square grid, $r$ is the maximum reach of a lamp, and $k$ is the number of lamps. The next $k$ lines will each contain two positive integers $i$ and $j$ ($1 \le i, j \le n$), indicating that there is a lamp in the grid at row $i$, column $j$.

Each lamp can illuminate squares at most $r$ units away, and can also illuminate its own square, so the maximum number of squares it can illuminate is $2 r + 1$. All lamps will be in distinct locations.

-----Output-----
Output a single integer, $1$ if it is possible to light all of the lamps and $0$ if it is not possible.

-----Examples-----
Sample Input:
3 2 5
1 1
1 3
3 1
3 3
2 2
Sample Output:
1
user: It is time again for the annual International Convention for Phonetic Communication. Because there are attendees from all over the world, and they do not all speak the same languages, the organizers have hired translators to help out.

To thank the translators at the end of the conference for their hard work, the organizers want to arrange a dinner at a nice local restaurant. However, the restaurant only has small, two person tables, hence the translators will have to be divided into pairs. As the organizers would like the translators to have a nice evening, they prefer that two translators sitting at the same table are both able to speak the same language. Write a program to help the organizers determine a way to match up the translators in pairs, so that each of the translators speaks a language that the other also speaks.

-----Input-----
The first line contains two numbers $N$ and $M$, the number of languages spoken at the convention, and the number of hired translators respectively ($2 \leq N \leq 100$, $1 \leq M \leq 200$).

The following $M$ lines each describe a translator. Each of these lines contains two integers specifying the two languages that the translator speaks. Languages are identified by integers in the range $[0,N-1]$.

Translators are identified by integers in the range $[0,M-1]$. Translators are listed in order of increasing identifier (i.e. the first listed translator has identifier $0$).

There are no two translators who both speak the same two languages. Translators have been chosen such that any two languages spoken at the conference can be translated into one another, although it may take several translators.

-----Output-----
If it is possible to match up all translators such that each pair speaks a common language, output a possible matching: print $M/2$ lines, each line containing the two identifiers of a pair of matched translators. The pairs, as well as the translators within a pair, may be listed in any order.

There may be multiple possible matchings. In that case, print any one of them.

If it is not possible to match up all translators, output a single line containing the word “impossible”.

-----Examples-----
Sample Input:
5 6
0 1
0 2
1 3
2 3
1 2
4 3
Sample Output:
5 3
1 0
2 4
user: A wireless sensor network consists of autonomous sensors scattered in an environment where they monitor conditions such as temperature, sound, and pressure. 

Samantha is a researcher working on the Amazon Carbon-dioxide Measurement (ACM) project. In this project, a wireless sensor network in the Amazon rainforest gathers environmental information. The Amazon rainforest stores an amount of carbon equivalent to a decade of global fossil fuel emissions, and it plays a crucial role in the world’s oxygen-transfer processes. Because of the huge size of this forest, changes in the forest affect not only the local environment but also global climate by altering wind and ocean current patterns. The goal of the ACM project is to help scientists better understand earth’s complex ecosystems and the impact of human activities.

Samantha has an important hypothesis and to test her hypothesis, she needs to find a subset of sensors in which each pair of sensors can communicate directly with each other. A sensor can communicate directly with any other sensor having distance at most $d$ from it. In order for her experiments to be as accurate as possible, Samantha wants to choose as many sensors as possible.

As one does not simply walk into the Amazon, Samantha cannot add new sensors or move those that are currently in place. So given the current locations of the sensors, she needs your help to find the largest subset satisfying her criteria. For simplicity, represent the location of each sensor as a point in a two-dimensional plane with the distance between two points being the usual Euclidean distance.

-----Input-----
The input consists of a single test case. The first line contains two integers $n$ and $d$ ($1 \le n \le 100$ and $1 \le d \le 10000$), where $n$ is the number of sensors available and $d$ is the maximum distance between sensors that can communicate directly. Sensors are numbered $1$ to $n$. Each of the next $n$ lines contains two integers $x$ and $y$ ($-10000\le x, y \le 10000$) indicating the sensor coordinates, starting with the first sensor.

-----Output-----
Display a maximum subset of sensors in which each pair of sensors can communicate directly. The first line of output should be the size of the subset. The second line of output should be the (one-based) indices of the sensors in the subset. If there are multiple such subsets, any one of them will be accepted.

-----Examples-----
Sample Input:
4 1
0 0
0 1
1 0
1 1
Sample Output:
2
1 2
user: Andy is going holiday shopping and needs to buy several Christmas trees. At the store, there is a row of Christmas trees of various heights. Andy is only allowed to buy trees that are next to each other, i.e. trees that are contiguous within the row. He also wants to make sure that the shortest tree he buys and the tallest tree he buys are as close in height as possible. What is the smallest height difference he can get given the constraints?

-----Input-----
The input consists of two lines. The first line contains two integers $n$ and $k$ ($2 \leq k \leq n \leq 200000$), representing the number of trees in the row and the number of trees Andy needs to buy, respectively. The second line consists of $n$ integers $a_1, a_2, \ldots , a_ n$ where $1 \leq a_ i \leq 100$, representing the height of each tree.

-----Output-----
Output the minimum height difference between the shortest and tallest trees of any contiguous subrange of $k$ trees.

-----Examples-----
Sample Input:
10 2
1 3 5 7 9 11 13 15 17 16
Sample Output:
1
user: Luka parked his truck near the lake. The lake is inhabited by the frog Barica, who jumps acrossplants floating on the lake’s surface. Knowing a fair number of folk tales, Luka knows that if he kisses Barica, she will turn into a beautiful princess. However, he needs to catch her first! Assuming an aerial view, the position of a plant on the lake’s surface can be defined with a paircoordinates. From plant $(x, y)$ Barica can jump:
 - To plant $(x+P, y+P)$, for any positive integer $P$. Call this direction $A$.
 - To plant $(x+P, y-P)$, for any positive integer $P$. Call this direction $B$.
 - To plant $(x-P, y+P)$, for any positive integer $P$. Call this direction $C$.
 - To plant $(x-P, y-P)$, for any positive integer $P$. Call this direction $D$.

Barica selects one of the four directions and jumps onto the first plant in the chosen direction. If there is no plant in the selected direction, Barica stays where she is. After Barica jumps, the plant she jumped from sinks and disappears.

Knowing the locations of the plants and the sequence of directions Barica chooses, Luka wantsdetermine coordinates of the plant Barica will end up on. Luka will wait for her at that plant, ambush her and kiss her.

Write a program that solves Luka’s problem and helps him turn Barica into a beautiful princess.

-----Input-----
The first line contains two integers $N$ and $K$ $(1 \leq N, K \leq 100000)$, the number of plants and the number of attempted jump. The second line contains $K$ letters each of which is ‘A’, ‘B’, ‘C’ or ‘D’. These letters represent in order the directions in which Barica attempts to jump.

Each of the following $N$ lines contains two integers $X$ and $Y$ $(0 \leq X \leq 1000000000, 0 \leq Y \leq 1000000000$), the coordinates of one plant. Barica is initially located on the first plant.

-----Output-----
Output Barica’s final coordinates.

-----Examples-----
Sample Input:
7 5
ACDBB
5 6
8 9
4 13
1 10
7 4
10 9
3 7
Sample Output:
7 4
user: From a pile of suggested tasks, authors of COCI must choose ones that will appear in the next round.

Difficulty of a task is described with an integer in range $1$ to $N$. For some tasks, however, it is not easy to exactly determine their difficulty. The COCI authors decided that these tasks can be considered as having either one of two consecutive difficulties. For example, some task can be considered as having difficulty of either $3$ or $4$.

The next round of COCI will contain exactly $N$ tasks. For each difficulty, there will be exactly one task with that difficulty. Of course, no task will appear twice.

Find the number of different ways authors can choose tasks for the next round. We say that two ways are different if for some difficulty, a different task is assigned to that difficulty.

Since the expected result can be very large, output the number of ways modulo $1000000007$.

-----Input-----
The first line of input contains the integer $N$ ($2 \le N \le 100000$).

The second line of input contains $N$ integers not greater than $10^9$. The $i$th number in this line is equal to the number of available tasks having difficulty exactly $i$.

The third line of input contains $N-1$ integers not greater than $10^9$. The $i$th number in this line is equal to the number of available tasks having difficulty either $i$ or $i+1$.

-----Output-----
The first and only line of output must contain the required number of ways modulo $1000000007$.

-----Examples-----
Sample Input:
3
3 0 1
0 1
Sample Output:
3
user: Once, two trees forgot their place and started to grow into each other. One of the trees grew from the left, and the other from the right. On $n$ points, they collided.

Numbering the points $1, 2, \dots , n$ from left to right, the left tree ended up connecting all of them in a single subtree rooted in node $1$, such that every node’s children had larger numbers than the node itself. We can describe this subtree with a list of $n-1$ edges.

Similarly, the right tree also connected all nodes in a single subtree rooted in node $n$, with every node’s children having smaller numbers than the node itself. This yields an additional $n-1$ edges.

Now, given the full list of $2(n-1)$ edges, it is not necessarily easy to tell which edge belongs to which tree. Can you figure out a possible assignment, or determine that it is impossible for this collection to have been the union of two trees?

-----Input-----
The first line of input contains the integer $n$ ($2 \le n \le 10^5$). The next $2(n-1)$ lines each contain two integers $u, v$ ($1 \le u < v \le n$) indicating an edge joining the two nodes $u$ and $v$. A pair $(u, v)$ may be connected by more than one edge.

-----Output-----
If it is possible for the edges to be the union of two trees that grow left-to-right and right-to-left, output a string of length $2(n-1)$, where the $i$’s character is L if the $i$’th edge should come from the left tree, or R if it should come from the right tree. Otherwise, output the word “impossible” on a single line. If there are multiple solutions, you may output any one of them.

-----Explanation of Sample Inputs-----
In the first example, there are two solutions: LLRRRRLL and LLRLRRLR.

In the second example, there are no solutions. Note that LRLR is not valid, because it would involve the right tree growing backward, from left to right.

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