meetforever/GRPO_CL huggingface dataset:
user: Rhymes are complicated. The extent to which one word rhymes with another depends on how similar they sound; but if they are too similar then they aren’t considered a rhyme at all. Karim has a huge list of $N$ words and wants to determine the maximally rhyming pair of words. To that end he has introduced the concept of rhyme power:

Given two words $S$ and $T$, their rhyme power $r(S,T)$ is defined as
 - $0$, if one word is a suffix of the other,
 - the length of their longest common suffix, otherwise.

For instance, the rhyme power of “fire” and “desire” is $3$, because their longest common suffix “ire” has length $3$. In contrast, the rhyme power of “impossible” and “possible” is $0$, because “possible” is a suffix of “impossible”.

Given Karim’s list of $N$ words, your task is to find the maximum value of $r(S, T)$ over all pairs of words.

-----Input-----
The first line contains an integer $N$ with $2 \leq N \leq 10^5$, the number of words.

The following $N$ lines each contain a string $S_ i$, where $1 \leq |S_ i| \leq 10^6$. The sum of all string lengths $|S_ i|$ is at most $10^6$.

-----Output-----
An integer, the maximal rhyme power.

-----Examples-----
Sample Input:
4
spaghetti
already
confetti
serengeti
Sample Output:
4
user: Nicholas Neverson was a student at Northlings Neverland Academy. As with any daydreaming student, Nicholas was playing around with a Kaleidoscope one day instead of paying attention to the teacher. Since this was math class, his daydreams quickly turned to palindromic numbers. A palindromic number is any number which reads the same forwards and backwards.

He describes his vision to you at lunch: numbers which are palindromic in several bases at once. Nicholas wonders how many such numbers exist. You decide you can quickly code up a program that given a range and a number $k$, outputs the number of numbers palindromic in all bases $j$, $2 \leq j \leq k$, in that range.

-----Input-----
Input consists of three space-separated integers: $a$, $b$, and $k$. The input satisfies the following constraints:\[ 0 \leq a \leq b \leq 2000000, \\ 2 \leq k \leq 100000. \]

-----Output-----
Output the quantity of numbers between $a$ and $b$ inclusive which are palindromes in every base $j$, for $2 \leq j \leq k$.

-----Examples-----
Sample Input:
1 356 2
Sample Output:
36
user: Sheila is a student and she drives a typical student car: it is old, slow, rusty, and falling apart. Recently, the needle on the speedometer fell off. She glued it back on, but she might have placed it at the wrong angle. Thus, when the speedometer reads $s$, her true speed is $s+c$, where $c$ is an unknown constant (possibly negative). 

Sheila made a careful record of a recent journey and wants to use this to compute $c$. The journey consisted of $n$ segments. In the $i^{\text {th}}$ segment she traveled a distance of $d_ i$ and the speedometer read $s_ i$ for the entire segment. This whole journey took time $t$. Help Sheila by computing $c$.

Note that while Sheila’s speedometer might have negative readings, her true speed was greater than zero for each segment of the journey.

-----Input-----
The first line of input contains two integers $n$ ($1 \le n \le 1000$), the number of sections in Sheila’s journey, and $t$ ($1 \le t \le 10^6$), the total time. This is followed by $n$ lines, each describing one segment of Sheila’s journey. The $i^{\text {th}}$ of these lines contains two integers $d_ i$ ($1 \le d_ i \le 1000$) and $s_ i$ ($|s_ i| \le 1000$), the distance and speedometer reading for the $i^{\text {th}}$ segment of the journey. Time is specified in hours, distance in miles, and speed in miles per hour.

-----Output-----
Display the constant $c$ in miles per hour. Your answer should have an absolute or relative error of less than $10^{-6}$.

-----Examples-----
Sample Input:
3 5
4 -1
4 0
10 3
Sample Output:
3.000000000
user: “[There] is a reason why the most successful trial lawyers are often master storytellers, making their cases come to life for their jurors.” ­–G. Christopher Ritter.

A good lawyer can take an incomplete set of facts and turn it into a convincing story.

Last night, a major robbery occurred at a bank, and, luckily, the robber seems to have been successfully arrested. Unfortunately, it is neither clear what the order of events was, nor whether the man they caught is even really the right guy. It is necessary to reconstruct the correct order of events to convince the jury that the accused is guilty beyond reasonable doubt for a conviction to be reached.

It is known that there are $N$ material events that are relevant to the case. For simplicity, we label the events $1, 2, \dots , N$. The prosecution claims that these events happened in the order $P_1, P_2, \dots , P_ N$, while the defense claims that these events instead happened in the order $D_1, D_2, \dots , D_ N$. It is guaranteed that exactly one of the two is right; in particular, this also means that the prosecution and defense do not claim the exact same order of events; that is, there exists some $i$ such that $P_ i \neq D_ i$.

Define a consistent recollection as a subset of the $N$ events, in some order, such that these events happening in this order is consistent with at least one of the two parties’ claims. (That is, at least one of the parties claimed that this subset of events did happen in this order, although it is possible that some other events happened before, after, or in between.) You need to determine
 - the minimum $k_ p$ such that there is some consistent recollection of $k_ p$ events where it is possible to determine for sure who is right, and
 - the minimum $k_ r$ such that for any consistent recollection of $k_ r$ events, it is possible to determine for sure who is right.

By finding these numbers, you can help the prosecution estimate the minimum amount of evidence they need to make their case, and the minimum amount of evidence they need to make their case foolproof. (Or determine that the man they have is really innocent.) Please help!

-----Input-----
The first line of input contains a single integer $N$ ($2 \leq N \leq 200000$), the number of events.

The second line of input contains $N$ integers, $P_1, P_2, \dots , P_ N$ ($1 \leq P_ i \leq N$; all $P_ i$ are distinct), the order of events according to the prosecution. That is, according to the prosecution, the $i^\text {th}$ event was the event labelled $P_ i$.

The third line of input contains $N$ integers, $D_1, D_2, \dots , D_ N$ ($1 \leq D_ i \leq N$; all $D_ i$ are distinct), the order of events according to the defense. That is, according to the defense, the $i^\text {th}$ event was the event labelled $D_ i$.

It is guaranteed that the prosecution and defense do not claim the exact same order of events; that is, there exists some $i$ such that $P_ i \neq D_ i$.

-----Output-----
Output two integers on a single line, the minimum values of $k_ p$ and $k_ r$, respectively, as described in the problem statement.

-----Examples-----
Sample Input:
4
3 2 4 1
1 3 4 2
Sample Output:
2 3
user: Lisa works as a waitress in a restaurant. Tonight is her birthday so Lisa asked the chef to prepare his special meal for her friends. The chef’s meal is made of $N$ ingredients. To prepare one serving of the meal he needs a certain amount of each ingredient.

There are some ingredients already available in the kitchen and Lisa will buy the rest at the grocery store. The store has all the necessary ingredients, each coming in smaller and larger packages. Lisa has $M$ dollars and wants to spend them so that the chef can make the most servings of his meal.

For example, consider the first sample data below. For $99$ dollars Lisa will buy three smaller and one larger package of the first ingredient, as well as one smaller and two larger packages of the second ingredient $(3\cdot 10 + 1 \cdot 11 + 1 \cdot 10 + 2 \cdot 24 = 99)$. The chef will then have $51$ units $(8 + 3 \cdot 10 + 1 \cdot 13)$ of the first ingredient and $60$ units $(20 + 1 \cdot 6 + 2 \cdot 17)$ of the second ingredient, enough for $5$ servings.

-----Input-----
The first line contains two integers $N$ and $M$, $1 \le N \le 100$, $1 \le M \le 100000$. Each of the following $N$ lines contains $6$ positive integers, information about one ingredient. These specify, in order:
 - $X$, $10 \le X \le 100$, the amount of the ingredient needed in one serving;
 - $Y$, $1 \le Y \le 100$, the amount of the ingredient already available in the kitchen;
 - $S_ M$, $1 \le S_ M < 100$, the size of the smaller package at the store;
 - $P_ M$, $10 \le PM < 100$, the price of the smaller package;
 - $S_ V$, $S_ M < S_ V \le 100$, the size of the larger package; and
 - $P_ V$, $P_ M < P_ V \le 100$, the price of the larger package.

-----Output-----
Output the largest number of servings the chef can make if Lisa spends her money wisely.

-----Examples-----
Sample Input:
2 100
10 8 10 10 13 11
12 20 6 10 17 24
Sample Output:
5
user: After Borko’s second plan of taking over the village fell through (because Mirko’s brother Stanko drank all the water), young Zvonko decided to relax with sports. For this purpose he stocked up on carrot juice and peanuts, and dived onto the couch, ready to watch the next tennis match on television.

While the two players are warming up, their statistics pop up on the screen. Zvonko noticed that the match history of the players contained invalid results. He had a brilliant idea where you write him a program that checks the validity of results, which he will then sell to the record keepers and enjoy a leisurely life.

A tennis match consists of sets, each set consisting of games. The following rules apply:
 - A player wins a set if he has $6$ or more games and at least two games more than his opponent.
 - Additionally, if the result is $6:6$ in the first or second set (but not the third set), a single final game is played to determine the winner of the set (the tie-break game).
 - The match ends when either player has won 2 sets. That player is the winner.

A match result is valid if a match could have been played by the above rules and ended in the result. Additionally, if one of the players is Roger Federer (designated as “federer” in the input), then a result in which he has lost a set can’t be valid (Zvonko knows Federer is from outer space).

Write a program that checks the validity of all matches between two players.

-----Input-----
The first line of input contains the names of the two players separated by a single space. Both names will be strings of at most $20$ lowercase letters of the English alphabet. The names will be different. The second line contains an integer $N$ $(1 \leq N \leq 50)$, how many matches the two players have played.

Each of the following $N$ lines contains the result of a single match, composed of the results of a number of sets separated by single spaces. Each match result will contain between $1$ and $5$ sets. A set is given in the format “$A$:$B$”, where $A$ and $B$ are games won by each player. These numbers are integers between $0$ and $99$ (inclusive).

-----Output-----
For each match, in the order they’re given in the input, output “da” if the result is valid, or “ne” if it isn’t.

-----Examples-----
Sample Input:
sampras agassi
6
6:2 6:4
3:6 7:5 2:6
6:5 7:4
7:6 7:6
6:2 3:6
6:2 1:6 6:8
Sample Output:
da
da
ne
da
ne
da
user: A new town is being planned, and the designers have some very specific ideas about how things should be laid out. First, they lay out the streets. Each street is perfectly straight and passes completely from one end of the town to the other. These streets divide the town into regions, and each region is to be designated either “residential” or “commercial.” The town planners require that any two regions directly across the street from one another must have different designations. On this one particular day, all of the streets have been planned, but none of the regions have been designated. One town planner wishes to purchase two properties, and it is important to him that the properties eventually have different designations. For this problem, the streets can be modeled by lines in the plane that extend forever in both directions and have no width, and properties may be modeled by points. Given the lines and two points, can you decide whether or not they must get different designations, “commercial” or “residential?” 

-----Input-----
Input begins with an integer $S$ on a single line, giving the number of streets ($1 \le S \le 10000$). The next $S$ lines of input each contain four integers $x_1$, $y_1$, $x_2$, and $y_2$, specifying the coordinates of two distinct points $(x_1, y_1)$ and $(x_2, y_2)$. The unique line through these two points gives one of the streets. Each coordinate is in the range $[0, 10000]$, and no two lines will be identical. That is, the town will have $S$ distinct streets. The next line contains an integer $T$, the number of pairs of properties to test ($ 1 \le T \le 1000$). This is followed by $T$ lines of input, each containing four integers $x_3$, $y_3$, $x_4$, and $y_4$, representing two distinct points $(x_3, y_3)$ and $(x_4, y_4)$, where each point lies within one of the two properties to test. None of these points will lie on any of the streets, nor will both points lie within the same property. Again, each coordinate is in the range $[0, 10000]$.

-----Output-----
For each of the $T$ pairs of properties to be tested, output either “same” if the properties are guaranteed to receive the same designation or “different” if they are guaranteed to receive different designations.

-----Examples-----
Sample Input:
2
1 1 2 1
1 1 1 2
3
2 0 2 2
2 0 0 3
0 0 2 2
Sample Output:
different
same
same
user: Oh no! Hackers are threatening to shut down Twitface, the premier social networking site. By taking advantage of lax security protocols, nefarious cyber-bandits have developed a virus that spreads from user to user, amplifying over time and eventually bringing the network to its knees from massive congestion. Normally users have to manually send messages to one another (squawking), but these ne’er-do-wells have figured out how to circumvent that rule, and have created squawks that spawn more squawks without user intervention. In particular, any time a user gets an infected squawk, one minute later it broadcasts an infected squawk to all its neighbors in the network (for purposes of this problem we assume that each neighbor gets the squawk exactly 1 minute after the initial user is infected). If a user receives multiple squawks at any point, the next minute it broadcasts that many squawks to all of its neighbors. For example, consider the following network:

If user $0$ is infected at time $t=0$, then at time $t=1$ users $1$ and $3$ get $1$ squawk each, at time $t=2$ users $0$ and $2$ get $2$ squawks each, and at time $t=3$, users $1$ and $3$ get $4$ squawks each and user $4$ gets $2$ squawks.

Given the layout of a social network and an initial infection site, you need to determine how many squawks are made at some given time $t$. In the above example the number of squawks would be 2, 4 and 10 at times 1, 2 and 3, respectively.

-----Input-----
The input will start with a line containing 4 integers $n$ $m$ $s$ $t$ indicating the number of users $(1\le n \leq 100)$, the number of links between users $(0\le m \leq n(n-1)/2)$, the index of the initially infected user $(s < n)$, and the number of minutes $(t < 10)$. Next will follow $m$ lines, each consisting of two integers $x$$y$, $(0 \leq x, y < n)$ indicating that users $x$ and $y$ are connected. Connections are symmetric and no two connections will be the same.



-----Output-----
Output the number of squawks sent at the specified time $t$.

-----Examples-----
Sample Input:
4 3 1 4
0 1
1 2
2 3
Sample Output:
8
user: Arthur and his sister Carol have been playing a game called Nim for some time now. Nim is played as follows:
 - The starting position has a number of heaps, all containing some, not necessarily equal, number of beads.
 - The players take turns chosing a heap and removing a positive number of beads from it.
 - The first player not able to make a move, loses.

Arthur and Carol really enjoyed playing this simple game until they recently learned an easy way to always be able to find the best move:
 - Xor the number of beads in the heaps in the current position (i.e., if we have $2$, $4$ and $7$ the xor-sum will be $1$ since $2\ \mathbf{xor}\ 4\ \mathbf{xor}\ 7 = 1$).
 - If the xor-sum is $0$, too bad, you will lose.
 - Otherwise, move such that the xor-sum becomes $0$. This is always possible.

It is quite easy to convince oneself that this works. Consider these facts:
 - The player that takes the last bead wins.
 - After the winning player’s last move the xor-sum will be $0$.
 - The xor-sum will change after every move.

Which means that if you make sure that the xor-sum always is $0$ when you have made your move, your opponent will never be able to win, and, thus, you will win.

Understandably it is no fun to play a game when both players know how to play perfectly (ignorance is bliss). Fortunately, Arthur and Carol soon came up with a similar game, $S$-Nim, that seemed to solve this problem. Each player is now only allowed to remove a number of beads in some predefined set $S$, e.g. if we have $S = \{ 2, 5\} $ each player is only allowed to remove $2$ or $5$ beads. Now it is not always possible to make the xor-sum $0$ and, thus, the strategy above is useless. Or is it?

Your job is to write a program that determines if a position of $S$-Nim is a losing or a winning position. A position is a winning position if there is at least one move to a losing position. A position is a losing position if there are no moves to a losing position. This means, as expected, that a position with no legal moves is a losing position.

-----Input-----
The first line of input contains a number $k$ ($1 \le k \le 100$) describing the size of $S$, followed by $k$ numbers $s_ i$ ($1 \le s_ i \le 10000$) describing $S$. The second line contains a number $m$ ($1 \le m \le 100$) describing the number of positions to evaluate. The next $m$ lines each contain a number $l$ ($1 \le l \le 100$) describing the number of heaps and $l$ numbers $h_ i$ ($0 \le h_ i \le 10000$) describing the number of beads in the heaps.

-----Output-----
For each position:
 - If the described position is a winning position print a ‘W’.
 - If the described position is a losing position print an ‘L’.

-----Examples-----
Sample Input:
2 2 5
3
2 5 12
3 2 4 7
4 2 3 7 12
Sample Output:
LWW
user: The small nation of Erratica prides itself on defying conventions established by the more “boring” countries around the world. One of their more obvious distinctions lies in the design of their coinage. Believing that a person should be easily able to identify the value a coin in a pocket or change purse by touch alone, Erratica designs its coins as polygons. For stability, the coins are convex – there are no notches cut into the coins. But the overall shapes themselves can be quite irregular.

Erratica Vending, the only manufacturer of vending machines in the nation, has been sent the design for a new coin to be issued. Their machines are designed so that coins enter through a slot into channel shaped as a rectangular prism with the slot being an open face of the prism. The channel is narrow because the coins are thin, but long enough to contain the entire coin. From the outside of the machine, the slot appears as a rectangular hole.

The company wants to know what would be the smallest slot size they will need so that the new coin can be slipped, after some rotation, into the slot.

-----Input-----
Input begins with a line containing $N$, the number of sides to the polygonal coin, where $3 \leq N \leq 20$.

This is followed by $N$ lines, each containing two real numbers $x$ and $y$, the coordinates of a vertex of the polygon, where $0 \leq x, y \leq 100$. All real numbers have at most $2$ digits after the decimal point.

All $N$ vertices will be distinct, and the vertices will be presented in an order proceeding clockwise around the perimeter of the coin.

-----Output-----
Print a single line with a real number denoting the minimum slot size allowing the coin to pass through. Outputs will be accepted that are within $\pm 0.01$ of the judges’ answer.

-----Examples-----
Sample Input:
3
0 0
0.71 3.54
4.21 4.21
Sample Output:
2.00111219
user: Your friend Tóti is an aspiring musician. He has written $n$ songs, each of which has a hype rating of either $1$, $2$, or $3$. A higher hype rating means the song has more energy. Tóti is planning his first live performance and needs your help. He wants to know how many setlists he can make. A setlist consist of at least three songs, the first song must have hype rating $1$, the last song must have hype rating $3$, and all other songs must have hype rating $2$. Tóti also wants to play the songs in the same order he wrote them. 

Given the hype rating of each of Tóti’s songs in the order he wrote them, how many setlists can he make?

-----Input-----
The first line of input consists of an integer $n$ ($1 \leq n \leq 10^6$), the number of songs Tóti has written. The second line consists of $n$ integers, each in $\{ 1, 2, 3\} $, giving the hype ratings of the $n$ songs in the order they were written.

-----Output-----
Output the number of setlists Tóti can make. Since this number can be large, print it modulo $10^9 + 7$.

-----Examples-----
Sample Input:
9
1 1 1 2 2 2 3 3 3
Sample Output:
63
user: Björn likes the square root of two, $\sqrt {2} = 1.41421356\dots $ very much. He likes it so much that he has decided to write down the first $10000$ digits of it on a single paper. He started doing this on an A4 paper, but ran out of space after writing down only $1250$ digits. Being pretty good at math, he quickly figured out that he needs an A1 paper to fit all the digits. Björn doesn’t have an A1 paper, but he has smaller papers which he can tape together.

Taping two A2 papers together along their long side turns them into an A1 paper, two A3 papers give an A2 paper, and so on. Given the number of papers of different sizes that Björn has, can you figure out how much tape he needs to make an A1 paper? Assume that the length of tape needed to join together two sheets of papers is equal to their long side. An A2 paper is $2^{-5/4}$ meters by $2^{-3/4}$ meters and each consecutive paper size (A3, A4, ) have the same shape but half the area of the previous one.

-----Input-----
The first line of input contains a single integer $2\leq n \leq 30$, the A-size of the smallest papers Björn has. The second line contains $n-1$ integers giving the number of sheets he has of each paper size starting with A2 and ending with A$n$. Björn doesn’t have more than $10^9$ sheets of any paper size.

-----Output-----
If Björn has enough paper to make an A1 paper, output a single floating point number, the smallest total length of tape needed in meters. Otherwise output “impossible”. The output number should have an absolute error of at most $10^{-5}$.

-----Examples-----
Sample Input:
4
1 0 5
Sample Output:
1.60965532263