meetforever/GRPO_CL huggingface dataset:
user: Jaap, Jan, and Thijs are on a trip to the desert after having attended the ACM ICPC World Finals 2015 in Morocco. The trip included a camel ride, and after returning from the ride, their guide invited them to a big camel race in the evening. The camels they rode will also participate and it is customary to bet on the results of the race. 

One of the most interesting bets involves guessing the complete order in which the camels will finish the race. This bet offers the biggest return on your money, since it is also the one that is the hardest to get right.

Jaap, Jan, and Thijs have already placed their bets, but the race will not start until an hour from now, so they are getting bored. They started wondering how many pairs of camels they have put in the same order. If camel $c$ is before camel $d$ on Jaap’s, Jan’s and Thijs’ bet, it means that all three of them put $c$ and $d$ in the same order. Can you help them to calculate the number of pairs of camels for which this happened?

-----Input-----
The input consists of:
 - one line with an integer $n$ ($2\leq n \leq 200000$), the number of camels;
 - one line with $n$ integers $a_1, \ldots , a_ n$ ($1 \le a_ i \le n$ for all $i$), Jaap’s bet. Here $a_1$ is the camel in the first position of Jaap’s bet, $a_2$ is the camel in the second position, and so on;
 - one line with Jan’s bet, in the same format as Jaap’s bet;
 - one line with Thijs’ bet, in the same format as Jaap’s bet.

The camels are numbered $1, \dots , n$. Each camel appears exactly once in each bet.

-----Output-----
Output the number of pairs of camels that appear in the same order in all $3$ bets.

-----Examples-----
Sample Input:
3
3 2 1
1 2 3
1 2 3
Sample Output:
0
user: Cars! Where do they come from? Where do they go? Nobody knows. They appear where roads have been built, as if out of nowhere. Some say that no two cars are alike. Some say that if you look closely, you can see the pale ghosts of miserable humans inside them, trapped forever—particularly in the morning and late afternoon. What scientific eye could frame their fearful symmetry?

Well, yours, hopefully. As part of your government’s Urban Traffic Control department, you are trying to write a paper on local traffic congestion. It is too dangerous to observe cars in the wild, of course, but you have been given some data on the traffic lights along your town’s Main Street, and you would like to do some theoretical calculations about how well-synchronized they are.

Main Street is set out on a line, with traffic lights placed at various points along it. Each traffic light cycles between red and green with a fixed period, being red for $r$ seconds, then green for $g$ seconds, then red for $r$ seconds, and so on. The values of $r$ and $g$ may be different for different traffic lights. At time $0$, all the lights have just turned red.

Assume that an “ideal” car mystically appears at the west end of Main Street at a uniformly random real-valued time in the interval $[0, 2019!]$ (where $k!$ is the product of the first $k$ positive integers), driving eastwards at a slow crawl of $1$ meter/second until it hits a red light. What is the probability that it will make it through all the lights without being forced to stop? If it does hit a red light, which one is it likely to hit first?

Write a program to answer these questions.

-----Input-----
The first line of input contains an integer $n$ ($1 \le n \le 500$), the number of traffic lights. Each of the following $n$ lines contains three integers $x$, $r$, and $g$ describing a traffic light, where $x$ ($1 \le x \le 10^5$) is the position of the light along Main Street in meters, and $r$ and $g$ ($0 \le r, g$ and $1 \le r+g \le 100$) are the durations in seconds of the red and green portions of the light’s period (so the light is red from time $0$ to $r$, from time $r+g$ to $2r+g$, and so on).

The west end of Main Street is at position $0$, and the lights are listed in order of strictly increasing position.

-----Output-----
For each of the $n$ lights, output a line containing the probability that this light will be the first red light an “ideal” car hits. Then output a line containing the probability that an “ideal” car makes it all the way without stopping. Your answers should have an absolute error of at most $10^{-6}$.

-----Examples-----
Sample Input:
4
1 2 3
6 2 3
10 2 3
16 3 4
Sample Output:
0.4
0
0.2
0.171428571429
0.228571428571
user: Ellen is teaching elementary math to her students and the time for the final exam has come. The exam consists of $n$ questions. In each question the students have to add ($+$), subtract ($-$) or multiply ($*$) a pair of numbers. 

Ellen has already chosen the $n$ pairs of numbers. All that remains is to decide for each pair which of the three possible operations the students should perform. To avoid students getting bored, Ellen wants to make sure that the $n$ correct answers to her exam are all different.

Please help Ellen finish constructing the exam by automating this task.

-----Input-----
The input consists of:
 - one line with one integer $n$ ($1\leq n \leq 2500$), the number of pairs of numbers;
 - $n$ lines each with two integers $a$ and $b$ ($-10^6 \leq a, b \leq 10^6$), a pair of numbers used.

-----Output-----
For each pair of numbers $(a,b)$ in the same order as in the input, output a line containing a valid equation. Each equation should consist of five parts: $a$, one of the three operators, $b$, an equals sign ($=$), and the result of the expression. All the $n$ expression results must be different.

If there are multiple valid answers you may output any of them. If there is no valid answer, output a single line with the string “impossible” instead.

-----Examples-----
Sample Input:
4
1 5
3 3
4 5
-1 -6
Sample Output:
1 + 5 = 6
3 * 3 = 9
4 - 5 = -1
-1 - -6 = 5
user: You’ve heard of acid rain but have you heard of antimatter rain? Antimatter rain is so potent that when it comes into contact with another object, it immediately disintegrates both itself and the object. Kayla’s job as a SpaceFleet Researcher is gathering weather data on exotic planets. This time, their assignment is to monitor the antimatter rainfall.

Sensors are set up in the planet’s atmosphere and are about to be rained on with antimatter rain. Oh no! Kayla monitors a single 2D section. Each sensor is either a single horizontal strip or a single point. When one or more antimatter droplet fall on a single sensor, all of those droplets and the sensor disintegrate simultaneously. That is, they disappear. All other droplets will drop past where the sensor used to be.

Kayla sees all the antimatter rain drops the moment before they all start to fall. All droplets fall at exactly the same rate.

For each droplet, Kayla wants to know if and where it will disintegrate. Help them out with this demanding task!

Illustration of the first sample. The vertical lines connect the drops to the sensor they hit. The drop with no associated vertical line will not hit any sensor.

-----Input-----
The first line of input contains two integers $D$ ($1 \leq D \leq 100000$), which is the number of antimatter droplets, and $S$ ($1 \leq S \leq 100000$), which is the number of sensors.

The next $D$ lines describe the droplets, in order. Each of these lines contains two integers $x$ ($1 \leq x \leq 10^9$), which is the $x$-coordinate of the droplet and $y$ ($1 \leq y \leq 10^9$), which is the $y$-coordinate of the droplet.

The next $S$ lines describe the sensors. Each line contains three integers $x_1$, $x_2$ ($1 \leq x_1 \leq x_2 \leq 10^9$), which is the leftmost and the rightmost $x$-coordinate of the sensor, and $y$ ($1 \leq y \leq 10^9$), which is the $y$-coordinate of the sensor.

It is guaranteed that no two drops will start in the same location, no drop will start on any sensor, and no two sensors touch (not even at a single point).

-----Output-----
For each droplet, in order, display a single number indicating the $y$-coordinate that it will disintegrate. If the droplet does not disintegrate, display $0$ instead. These values should appear on separate lines.

-----Examples-----
Sample Input:
5 3
1 8
2 3
2 8
5 8
5 9
3 6 6
1 7 4
1 3 1
Sample Output:
4
1
4
6
0
user: Given $N$ integers in the range $[-50000, 50000]$, how many ways are there to pick three integers $a_ i$, $a_ j$, $a_ k$, such that $i$, $j$, $k$ are pairwise distinct and $a_ i + a_ j = a_ k$? Two ways are different if their ordered triples $(i, j, k)$ of indices are different.

-----Input-----
The first line of input consists of a single integer $N$ ($1 \leq N \leq 200000$). The next line consists of $N$ space-separated integers $a_1, a_2, \dots , a_ N$.

-----Output-----
Output an integer representing the number of ways.

-----Examples-----
Sample Input:
4
1 2 3 4
Sample Output:
4
user: One of the perhaps most important factors in deciding the winning team of a football match is the amount of cheers the teams get from their supporters. The Sportify football team is about to meet their worst opponent, Spoilify. As the manager of Sportify’s cheerleading squad, you have to decide on a cheerleading tactic which will ensure you the win.

At your disposal you have $n$ cheerleaders. Each cheerleader has energy to cheer for a total of $t$ minutes of the 90 minute long game (the cheerleaders do not recover energy during the short half-time intermission). For each cheerleader you have to decide when in the game they should cheer. The cheerleading schedule for a match is divided into one minute intervals, and during each interval each cheerleader either cheers or does not cheer during the entire interval.

If a team gets more cheers than the other team for 5 consecutive minutes of the game, that team will score a goal. The 5 minute count is then reset, so if a team gets more cheers than the other team for, say, 17 consecutive minutes, that team will score three goals. The 5 minute interval can not run over the half time intermission, e.g., getting more cheers than the other team from the 43’rd minute to the 49’th minute is not sufficient to score a goal.

As an additional ace up your sleeve, you have spied on the Spoilify cheerleading team and know what their strategy for the game is. Based on this information, what is the best result that can be obtained?

-----Input-----
The first line of input consists of three integers $0 \le n \le 20$, $1 \le t \le 90$, and $0 \le m \le 1000$. The meanings of $n$ and $t$ are described above, and $m$ gives the number of time intervals that the Spoilify cheearleading team is active. Next follow $m$ lines, each consisting of two integers $a$ and $b$, $0 \le a < b \le 90$, indicating that a cheerleader from Spoilify is cheering from time $a$ to time $b$ of the match. These intervals may overlap or even be equal, meaning that several cheerleaders are active simultaneously.

-----Output-----
Output two integers $A$ and $B$ where $A$ is the number of goals scored by the Sportify team and $B$ is the number of goals scored by the Spoilify team, assuming you use an optimal cheerleading tactic. If there are many possible results, you should first attempt to maximize the score difference $A-B$, and if there are still many possible results, maximize your score $A$.

-----Examples-----
Sample Input:
1 31 2
20 60
50 90
Sample Output:
4 3
user: The host university is organizing a party for this year’s ACM/ICPC contestants with a buffet dinner and $R$ boxes of red wine and $W$ boxes of white wine. Wine boxes are to be arranged into non-empty piles, each pile contains only one type of wine, either white or red. After that, those piles are put into a line so that no two piles belonging to the same kind sit next to each other. In addition, for security reasons, each red wine pile should not have more than $d$ boxes of red wine, although piles of white wine can be arbitrarily large.

Your task is to identify $K$ - the number of different ways to arrange the given wine boxes satisfying all of the above conditions.

-----Input-----
The input contains $3$ space separated integers in one line: $R$, $W$, $d$ $(1 \leq R,W \leq 10^6, 1 \leq d \leq R)$

-----Output-----
Write in one line the remainder of $K$ divided by $10^9+7$.

-----Sample Clarification-----
In the first sample below, there are $3$ valid arrangements:

In the second sample below, there are $6$ valid arrangements:

-----Examples-----
Sample Input:
2 2 1
Sample Output:
3
user: Frustrated by his legendary inability to tear down Lund’s Cathedral, Finn the Giant wants to become better at destroying pillar-borne edifices. Together with his fellow trolls he has constructed a training site consisting of a colonnade of various pillars supporting a roof. The pillars differ in structural strength—some are hewn of Swedish granite, but in various styles, others are made of stone, clay, or even wood. The roof his a huge block of uniform granite. You can assume that in the beginning, each pillar is able to support its own part of the roof. The pillars have exactly the same distance.

As an experienced builder, Finns known enough structural engineering to understand how roof weight distributes across pillars: every pillar supports the part of the roof closest to it. After removing pillars 1 and 2, the middle pillar supports the shaded part of the roof:

At the left of pillar 0 and to the right of pillar $n-1$, the roof is supported by two indestructible pillars. Initially, every pillar is subject to the same pressure, 1000 kN.

Finn is able to tear down exactly one of the internal pillars, independent of its structural integrity. He hopes to start a chain reaction, whereby the redistribution of the roof’s weights results in other pillars being crushed. Which pillar should Finn tear down to create the maximum damage?

-----Input-----
Input consists of two lines. On the first row, the number $n$ of pillars, with $1\leq n\leq 100000$. On the second row, $n$ integers $b_0$, $\ldots $, $b_{n-1}$, where $1000 \leq b_ i\leq 100000000$, representing how much each internal pillar can support (in kN), from left to right.

-----Output-----
A single line with two integers. First the maximal damage (counted in number of destroyed pillars) that Finn can cause. Second, the pillar he has to tear down. The internal pillars are numbered $0$, $1$, $\ldots $, $n-1$. If there are more than one correct answers, any will do.

-----Examples-----
Sample Input:
5
1341 2412 1200 3112 2391
Sample Output:
3 1
user: Alice and Bob are playing a board game. The board is divided into positions labeled $a, b, c, d, \dots $ and the players use a gamepiece to mark the current position. Each round of the game consists of two steps:
 - Alice makes a choice. Depending on the current position, she has different options, where each option is a set of positions. Alice chooses one set $S$ among the available sets of positions.
 - Bob makes a choice. His choice is one position $p$ from the set $S$ that Alice chose in step 1. Bob moves the gamepiece to position $p$, which is the position for the start of the next round.

Prior to the first round, each player independently selects one of the positions and reveals it at the start of the game. Bob’s position is where the game starts. Alice wins the game if she can force Bob to move the gamepiece to the position she has chosen. To make things interesting, they have decided that Bob will pay Alice a certain amount if he loses, but Alice must pay Bob a certain amount after every round. The game now ends if Alice’s position is reached or when Alice runs out of cash.

Both Alice and Bob play optimally: Alice will always choose an option that will lead to her winning the game, if this is possible, and Bob will always try to prevent Alice from winning.

For all possible start and end positions, Alice would like you to determine whether she can win the game and if so, how many rounds it will take.

-----Input-----
The input consists of a single test case. The first line contains the number of positions $n$ ($1 \leq n \leq 25$). The $n$ positions are labeled using the first $n$ letters of the English alphabet in lowercase. The rest of the test case consists of $n$ lines, one for each position $p$, in alphabetical order. The line for position $p$ contains the options available to Alice in position $p$. It starts with the number of options $m$ ($1 \leq m < 2^n$), which is followed by $m$ distinct strings, one for each option. Each string contains the positions available to Bob if Alice chooses that option. The string has at least $1$ character, the characters (which correspond to valid board positions) are in alphabetical order, and no characters are duplicated. The total number of options for the test case is at most $10^6$.

-----Output-----
For each position $p$ in alphabetical order, display one line. In that line, for each position $q$ in alphabetical order display the minimal number of rounds in which Alice can be guaranteed to arrive at position $q$ when starting the game in position $p$, or $-1$ if Alice cannot be guaranteed to reach $q$ from $p$.

-----Examples-----
Sample Input:
2
2 ab b
1 b
Sample Output:
0 1 
-1 0
user: The game Pokenom Go has just been released. Pokenom trainers can now travel the world, capture Pokenom in the wild and battle each other! Bash — the Pokenom trainer — has decided to drop out of his university to pursue his childhood dream of becoming the best Pokenom trainer!

However, Linux — Bash’s university headmaster — does not allow his students to drop out so easily …

Linux puts $N$ black boxes on a straight line. The black boxes are numbered from $1$ to $N$ from left to right. Initially, all black boxes are empty. Then Linux gives Bash $Q$ queries. Each query can be one of the following $2$ types:
 - Linux puts exactly one stone inside exactly one box between $u$-th box and $v$-th box, inclusive, with equal probability. $(1 \le u \le v \le N)$.
 - Let $a_ i$ be the number of stones in black box numbered $i$. Let $A = \sum _{i=1}^{N}{a_ i^2}$. Bash has to calculate the expected value $E(A)$.

Bash can only drop out of his university if he is able to answer all queries correctly. But now all Bash can think of is Pokenom. Please help him!

-----Input-----
The first line of input contains exactly $2$ positive integers $N$ and $Q$. $(1 \le N, Q \le 10^5)$.

$Q$ lines follow, each line contains exactly one query. As explained, a query can be one of the following $2$ types:
 - $1 \; u \; v$: Linux puts a stone inside one of the boxes between $u$ and $v$.
 - $2$: Linux asks Bash to compute $E(A)$.

-----Output-----
It can be proved that the expected value can be represented as an irreducible fraction $\dfrac {A}{B}$. For each query of type $2$, print one line containing the value $A \times B^{-1}$ modulo $10^{9} + 7$. The given input guarantees that $B$ is not a multiple of $10^{9} + 7$.

-----Explanation for examples-----
 - In the first example: With a probability of $0.5$, two stones are in different squares. Hence, the answer to the fourth query is $0.5 \times (1^{2} + 1^{2}) + 0.5 \times 2^{2} = 3$.
 - In the second example: With a probability of $\frac{2}{3}$, two stones are in different squares. Hence, the answer to the fourth query is $\frac{2}{3} \times 2 + \frac{1}{3} \times 4 = \frac{8}{3}$.

-----Examples-----
Sample Input:
2 4
1 1 2
2
1 1 2
2
Sample Output:
1
3
user: Chances are that you have probably already heard of the travelling salesman problem. If you have, then you are aware that it is an NP-hard problem because it lacks an efficient solution. Well, this task is an uncommon version of the famous problem! Its uncommonness derives from the fact that this version is, actually, solvable.

The travelling salesman is on a mission to visit $N$ cities, each exactly once. The cities are represented by numbers $1, 2, \ldots , N$. What we know is the direct flight duration between each pair of cities. The salesman, being the efficient man that he is, wants to modify the city visiting sequence so that the total flight duration is the minimum possible.

Alas, all is not so simple. In addition, the salesman has a peculiar condition regarding the sequence. For each city labeled $K$ the following must apply: either all cities with labels smaller than $K$ have been visited before the city labeled $K$ or they will all be visited after the city labeled $K$. In other words, the situation when one of such cities is visited before, and the other after is not allowed.

Assist the poor fellow in his ambitious mission and calculate the minimum total flight duration needed in order to travel to all the cities, starting from whichever and ending in whichever city, visiting every city exactly once, so that his peculiar request is fulfilled.

-----Input-----
The first line of input contains the positive integer $N$ ($2 \leq N \leq 1500$), the number of cities. Each of the following $N$ lines contains $N$ positive integers from the interval $[0,1000]$. The number in $B$-th place in the $A$-th row represents the flight duration between cities $A$ and $B$; that number is equal to the $A$-th number in the $B$-th row. When $A = B$, that number is 0. Otherwise, it is a positive value.

-----Output-----
The first and only line of output must contain the required minimum total flight duration.

-----Examples-----
Sample Input:
3
0 5 2
5 0 4
2 4 0
Sample Output:
7
user: You are playing the following simple game with a friend:
 - The first player picks a positive integer $X$.
 - The second player gives a list of $k$ distinct positive integers $Y_1, \ldots , Y_ k$ such that $(Y_1+1)(Y_2+1) \cdots (Y_ k+1) = X$, and gets $k$ points.

Write a program that plays the second player.

-----Input-----
The input consists of a single integer $X$ satisfying $10^3 \le X \le 10^{15}$, giving the number picked by the first player.

-----Output-----
Write a single integer $k$, giving the number of points obtained by the second player, assuming she plays as good as possible.

-----Examples-----
Sample Input:
1099511627776
Sample Output:
8