meetforever/GRPO_CL huggingface dataset:
user: “In the casino, the cardinal rule is to keep them playing and to keep them coming back. The longer they play, the more they lose, and in the end, we get it all.”  (from the 1995 film Casino) 

Recent recessions have not been kind to entertainment venues, including the gambling industry. Competition is fierce among casinos to attract players with lots of money, and some have begun to offer especially sweet deals. One casino is offering the following: you can gamble as much as you want at the casino. After you are finished, if you are down by any amount from when you started, the casino will refund $x\% $ of your losses to you. Obviously, if you are ahead, you can keep all of your winnings. There is no time limit or money limit on this offer, but you can redeem it only once.

For simplicity, assume all bets cost 1 dollar and pay out 2 dollars. Now suppose $x$ is 20. If you make 10 bets in total before quitting and only 3 of them pay out, your total loss is 3.2 dollars. If 6 of them pay out, you have gained 2 dollars.

Given $x$ and the percentage probability $p$ of winning any individual bet, write a program to determine the maximum expected profit you can make from betting at this casino, using any gambling strategy.

-----Input-----
The input consists of a single test case. A test case consists of the refund percentage $x$ ($0 \le x < 100$) followed by the winning probability percentage $p$ ($0 \le p < 50$). Both $x$ and $p$ have at most two digits after the decimal point.

-----Output-----
Display the maximum expected profit with an absolute error of at most $10^{-3}$.

-----Examples-----
Sample Input:
0 49.9
Sample Output:
0.0
user: The people of Absurdistan discovered how to build roads only last year. After the discovery, each city decided to build its own road, connecting the city to some other city. Each newly built road can be used in both directions.

You bought a tourist guide which has a map of the country with the newly built roads. However, since you are very interested in history, you would like to know which city built which road.

Given the description of $n$ roads, can you find an assignment of roads to $n$ cities, such that each city built one road? If there are multiple assignments, you are happy with any one. At least one solution is guaranteed to exist.

-----Input-----
The first line contains an integer $n$ $(2\le n\le 100000)$ – the number of cities and roads. Then follow $n$ lines with $2$ numbers each. A line containing “$a$ $b$” indicates that there is a road between cities $a$ and $b$, $1 \le a, b\le n, a \not= b$. There can be multiple roads between the same pair of cities.

-----Output-----
Print $n$ lines with two integers “$a$$b$” denoting that a road between $a$ and $b$ was built by city $a$. Each road from the input must appear exactly once in the output. If there are multiple solutions, you can print any one and you can print the roads in any order.

-----Examples-----
Sample Input:
4
1 2
2 3
3 1
4 1
Sample Output:
4 1
2 1
3 2
1 3
user: Every day, Danny buys one sweet from the candy store and eats it. The store has $m$ types of sweets, numbered from $1$ to $m$. Danny knows that a balanced diet is important and is applying this concept to his sweet purchasing. To each sweet type $i$, he has assigned a target fraction, which is a real number $f_ i$ ($0 < f_ i \le 1$). He wants the fraction of sweets of type $i$ among all sweets he has eaten to be roughly equal to $f_ i$. To be more precise, let $s_ i$ denote the number of sweets of type $i$ that Danny has eaten, and let $n = \sum _{i=1}^m s_ i$. We say the set of sweets is balanced if for every $i$ we have\[ n f_ i - 1 < s_ i < n f_ i + 1. \]

Danny has been buying and eating sweets for a while and during this entire time the set of sweets has been balanced. He is now wondering how many more sweets he can buy while still fulfilling this condition. Given the target fractions $f_ i$ and the sequence of sweets he has eaten so far, determine how many more sweets he can buy and eat so that at any time the set of sweets is balanced.

-----Input-----
The input consists of three lines. The first line contains two integers $m$ ($1 \le m \le 10^5$), which is the number of types of sweets, and $k$ ($0 \le k \le 10^5$), which is the number of sweets Danny has already eaten.

The second line contains $m$ positive integers $a_1, \ldots , a_ m$. These numbers are proportional to $f_1, \ldots , f_ m$, that is, $\displaystyle f_ i = \frac{a_ i}{\sum _{j = 1}^m a_ j}$. It is guaranteed that the sum of all $a_ j$ is no larger than $10^5$.

The third line contains $k$ integers $b_1, \ldots , b_ k$ ($1 \le b_ i \le m$), where each $b_ i$ denotes the type of sweet Danny bought and ate on the $i^\text {th}$ day. It is guaranteed that every prefix of this sequence (including the whole sequence) is balanced.

-----Output-----
Display the maximum number of additional sweets that Danny can buy and eat while keeping his diet continuously balanced. If there is no upper limit on the number of sweets, display the word forever.

-----Examples-----
Sample Input:
6 5
2 1 6 3 5 3
1 2 5 3 5
Sample Output:
1
user: You are a very busy person, with a lot of important meetings. Today, you have a meeting for which it is insanely important to arrive at the agreed time. 

Luckily you live in Zürich, which features a large network of extremely punctual trams. Each tram line travels from one place to another at regular intervals, always taking the same time from departure to arrival. It is very easy to change trams, and we assume that it takes no time to change to another tram if both are at a stop at the same time. This means that if a tram arrives at its destination at exactly time $t$ and another tram departs from the same place at time $t$ (or later), you will have enough time to change tram.

You are currently working in your hotel room before the meeting. Since you are a very busy person, you would like to leave your hotel at the latest possible time possible while still ariving to the meeting on time. When do you need to leave for your meeting?

-----Input-----
The input consists of:
 - one line with three integers $n$, $m$ and $s$ ($2 \le n \leq 100000$, $1 \le m \leq 200000$, $1 \le s \leq 10^9$), the number of tram stops, the number of tram lines, and the time at which the meeting starts in seconds relative to now.
 - $m$ lines, each with five integers $u, v, t_0, p, d$ ($0 \le u \not= v < n$, $0 \le t_0 \le 10^9$, $1 \le p, d \le 10^9$). The $i$’th line describes the $i$’th tram line, which departs from tram stop $u$, arrives at tram stop $v$, starts its first departure $t_0$ seconds from now, departs every $p$ seconds from the first departure, and takes $d$ seconds from departure to arrival.

The stops are numbered between $0$ and $n - 1$. Your hotel is located at stop $0$, and the meeting is at stop $n - 1$.

-----Output-----
Output the latest time at which you can leave the hotel while arriving to your meeting on time, in seconds from now. If you can not make it to your meeting on time, output impossible instead.

-----Examples-----
Sample Input:
2 1 10
0 1 1 2 6
Sample Output:
3
user: Sandra recently bought her first smart phone. One of her friends suggested a long list of applications (more commonly known as “apps”) that she should install on the phone. Sandra immediately started installing the apps from the list, but after installing a few, the phone did not have enough disk space to install any more apps. Sometimes, the app installation failed because there was not even enough space to download the installation package. Other apps could be downloaded just fine, but had insufficient space to store the installed app. 

Each app that Sandra installs has a download size $d$ and a storage size $s$. To download the app, Sandra’s phone must have at least $d$ megabytes of free disk space. After the app has been installed, it then uses $s$ megabytes of disk space on the phone. The download size may be smaller than the storage size (e.g., if the app data is heavily compressed) or larger than the storage size (e.g., if the download contains material that might not get used such as translations to different languages). The installer is very efficient and can transform the downloaded package to an installed app without using any extra disk space. Thus, to install an app, the phone must have at least $\max (d, s)$ megabytes of free disk space.

Sandra quickly realised that she may have run out of space just because she installed apps in the wrong order. Thus, she decided to give the installation another try. She uninstalled all apps, and will now choose an installation order that lets her install the largest number of apps from the list. Sandra may not install any app more than once.

Help her determine what apps on the list she should install, and in what order.

-----Input-----
The input consists of:
 - One line with two integers $n$, $c$ ($1 \le n \le 500, 1 \le c \le 10000$), the number of available apps and the available disk space of the phone in megabytes.
 - $n$ lines, each with two integers $d, s$ ($1 \le d, s \le 10000$), the download size and storage size of an app, in megabytes.

-----Output-----
Output one line with the maximum number of apps that can be installed. Then output one line listing the numbers of those apps, in the order that Sandra should install them. In the case that no apps can be installed, this line can be omitted.

The apps are numbered from $1$ to $n$, in the order they are given in the input. If there are multiple optimal solutions, output any one of them.

-----Examples-----
Sample Input:
2 100
99 1
1 99
Sample Output:
2
1 2
user: As a Hunter, you will undoubtedly face difficult obstacles during your journey. At such time, a Hunter is expected to model the situation using Mathematical models, and apply probability and statistics knowledge to overcome the situation.

Thus, the third round of the Hunter Exam is based on the following game with coins:
 - Before the game starts, Gon selects a string $g$ and Killua selects a string $k$. The two strings must contain only characters ‘H’ and ‘T’.
 - A game master will flip a coin an infinite number of times. After each flip, the result (either ‘H’ or ‘T’ — representing head or tail) is appended into a string $s$. The string $s$ is empty at the beginning of the game.
 - After some coin flip:
 - If both $g$ and $k$ become a substring of $s$, the game ends in a draw.
 - If only $g$ becomes a substring of $s$, Gon wins, and the game ends.
 - If only $k$ becomes a substring of $s$, Killua wins, and the game ends.
 - Gon and Killua only have finite amount of time. They will stop the game in a draw after $10^{100}$ turns.

Please calculate the probability that Gon wins.

-----Input-----
The input contains exactly three lines:
 - The first line contains the string $g$.
 - The second line contains the string $k$.
 - The third line contains a real number $p$ with exactly one digit after the decimal point — the probability that a coin flip will result in head $(0 < p < 1)$.

The length of $g$ and $k$ are at most $20$. $g$ and $k$ are non-empty, and contain only characters ‘H’ and ‘T’.

-----Output-----
The output must contain a single number — the probability that Gon wins.

Your answer will be considered correct if its relative or absolute error doesn’t exceed $10^{-6}$.

Namely: let’s assume that your answer is $a$, and the answer of the jury is $b$. The checker program will consider your answer correct, if $\frac{|a-b|}{max(1,b)} \leq 10^{-6}$.

-----Examples-----
Sample Input:
H
T
0.5
Sample Output:
0.5
user: During a game of Minesweeper you have arrived in the situation shown in the image, with an $n\times n$ square surrounded by $1$s, themselves surrounded by unknown cells. Your task is to determine which of the $4n+4$ surrounding $1$-cells is guaranteed to not contain a mine. 

If you are unfamiliar with the rules of Minesweeper, here is a description:
 - Each of the red-labelled cells is either empty or contains a mine. All other cells are empty.
 - Each cell with a blue $1$ is adjacent to exactly one mine. Two cells are said to be adjacent if they share an edge or a corner. Thus, except at the border, every cell is adjacent to $8$ other cells.

Determine which of the red-labelled cells are safe, i.e., guaranteed to not contain a mine.

-----Input-----
An integer $n$ with $1 \leq n \leq 1000$, the side length of the square. The image corresponds to $n = 6$.

It can be shown that there exists at least one valid placement of mines for each $n$.

-----Output-----
First print the number $m$ of safe cells. Then print a line with $m$ integers, the indices of the safe cells in increasing order. The cells are indexed clockwise from $1$ to $4n+4$, starting at the bottom left corner, as shown in the image.

-----Examples-----
Sample Input:
3
Sample Output:
8
1 3 5 7 9 11 13 15
user: Due to the frantical usage of the racket to kill flies, Marin has sustained a serious bodily injury known to the medical community as epicondylitis lateralis humeri. His grandma has advised smearing rakija over it, the doctor has prescribed a strong painkiller, but Marin has ignored every single advice and decided to look for the answer in integer sequences.

He has discovered a previously undiscovered sequence of integers and called it the xorbonacci sequence. The $n$-th element in the sequence is denoted with $x_ n$. The sequence is defined recursively in the following way:\begin{align*} x_1 & = a_1,\\ x_2 & = a_2,\\ & \ldots \\ x_ k & = a_ k,\\ x_ n & = x_{n-1} \oplus x_{n-2} \oplus \ldots \oplus x_{n-k}, \quad n > k\\ \end{align*}

Because of a reason only known to Marin, he determined that all his sorrows will go away if you answer his $Q$ queries defined with numbers $l$ and $r$. The answer to the query is represented with the value\begin{equation*} x_ l \oplus x_{l+1} \oplus \ldots \oplus x_{r-1} \oplus x_ r \end{equation*}

Help Marin and answer his queries.

Please note: The operation $\oplus $ is the operation of binary XOR.

-----Input-----
The first line of input contains the integer $K$ ($1 \leq K \leq 100000$) from the task. The following line contains $K$ integers that represent the first $K$ elements in the xorbonacci sequence. All numbers are smaller than $10^{18}$. The following line contains the integer $Q$ ($1 \leq Q \leq 10^6$) from the task. The $i$-th of the following $Q$ lines contains two integers $l_ i$ and $r_ i$ ($1 \leq l_ i \leq r_ i \leq 10^{18}$) that represent Marin’s $i$-th query.

-----Output-----
Each of the following $Q$ lines of output must contain the answers to Marin’s queries, the order being the same as the input.

-----Examples-----
Sample Input:
4
1 3 5 7
3
2 2
2 5
1 5
Sample Output:
3
1
0
user: Will the galaxy see peace at last? All the warlords have gathered to divide all of space between themselves. The negotiations have come quite far and the warlords have finally agreed on a peaceful way of deciding who gets what. The $2$-dimensional galactic map must first be divided into sectors by splitting it along a set of infinite lines. The warlord with the largest battle fleet will choose one sector, then the warlord with the second largest fleet will choose some other sector and so on, until everyone has gotten a sector. This is then repeated until there are no sectors left.

Different sets of lines have been suggested, and it is up to you to present these alternatives to the meeting. To make sure that there will be peace, you are ready to modify the suggestions slightly. You have some experience with warlords and know that no warlord will settle for less space than anyone else, so for there to be peace, all of them must get the exact same area on the map. Since space is infinite, so is the map. Some sectors therefore have infinite area, so that is the amount of space everyone wants. How many extra lines do you have to add to make sure each warlord can get at least one sector with infinite area?

-----Input-----
The first line of input contains two positive integers $W$ and $N$, ($1 \leq W,N \leq 100$) denoting the number of warlords and the number of lines in the suggested division of space. This is followed by $N$ lines each containing four integers $x_1$, $y_1$, $x_2$ and $y_2$, each with an absolute value no higher than $10000$. This means that one line is intersecting the two points $(x_1,y_1)$ and $(x_2,y_2)$ on the galactic map. These two points are not be the same.

-----Output-----
Output the number of lines you have to add to this suggestion to satisfy all warlords.

-----Examples-----
Sample Input:
2 1
1 1 -2 0
Sample Output:
0
user: A well-known Canadian programming contest is always held on a Friday in October each year. We need you to help choose the dates for years $2019$, $2020$, …, $2018+Z$.

There are Fridays when we cannot schedule the contest because of conflicting events. In particular, the contest cannot be held on the Friday before Canadian Thanksgiving; that date is sacred. A list of additional “forbidden dates” is provided, but every year there is at least one Friday in October when the contest can be held.

Not all contest schedules are equally good. People tend to expect the contest to be held on or near the date it was held in the previous year. For example, they don’t like to be surprised when they expect a date in early October but the contest runs late in the month, etc. There is a yearly “surprise penalty” for holding the contest on October $X$ one year when it had been on October $Y$ the year before, namely $(X-Y)^2$, which penalizes big surprises much more than little ones.

Your goal is to find a schedule for the $Z$ years following $2018$ such that the total of the yearly surprise penalties is as small as possible.Useful facts:
 - Canadian Thanksgiving is always the second Monday in October.
 - January $1$, $2019$, is a Tuesday.
 - Years have $365$ days, except leap years, which have an extra day in February.
 - From $2018$ until the year $2400$ (exclusive), a leap year is any year divisible by $4$ but not divisible by $100$.
 - January $1$ is the $1^{\textrm{st}}$ day of the year, and October $1$ is the $274^{\textrm{th}}$ day of $2018$.
 - October has $31$ days.
 - The contest was held on October $12$, $2018$.

-----Input-----
 - The first line of the input is the integer $Z$ $(1 \leq Z \leq 100)$, the number of years following $2018$ for which the contest is to be scheduled.
 - The second line contains an integer $F$ $(0 \leq F \leq 50)$, the number of forbidden dates.
 - The following $F$ lines contain the forbidden dates, one per line. Each forbidden date is represented by three space-separated numbers:
 - The first number consists of $4$ digits and represents the year. It is between $2019$ and $2118$, inclusive.
 - The second number is always $10$, representing October.
 - The third number consists of two digits and represents the day in October. It ranges between $01$ and $31$, inclusive.

The dates are given in chronologically increasing order. There is no guarantee that all forbidden dates are Fridays.

-----Output-----
The first line of the output is an integer representing the smallest total surprise penalty that any schedule can achieve. The remaining $Z$ lines contain the chosen dates of a schedule with this total surprise penalty, one date per line, in chronologically increasing order. Note that the input data is chosen so that this schedule is unique. Dates should be formatted as in the input: $4$ digits, a space, the number $10$, a space, and $2$ digits.

-----Examples-----
Sample Input:
2
5
2019 10 18
2019 10 19
2020 10 02
2020 10 16
2020 10 23
Sample Output:
194
2019 10 25
2020 10 30
user: You are in a game of laser tag, where the goal is to shoot and hit other players with the laser gun you are carrying. Running up to them and shooting sometimes works, but it leaves you more vulnerable to their counterattack than you would like. You have found that you can be sneakier if you bounce your shot off of a mirror before it hits them. This allows you to shoot around a corner, for example.

To practice your shots, you shoot at a wall (the target) by reflecting your shot off of a mirror. Write a program that tells you where on the target wall your laser would strike when shooting from different locations. The wall is considered to be infinitely long at $x=0$.

-----Input-----
Input consists of up to $1500$ test cases. Each test case consists of six space-separated real values, $x_1~ y_1~ x_2~ y_2~ x_3~ y_3$. The first four numbers represent the endpoints of the mirror, and the last two represent your position. You can be sure that $x_1 \neq x_2$, or $y_1 \neq y_2$, or both. You can also be sure that $x_1\cdot x_2>0$. You are not standing on the mirror. All coordinates are in the range $[-100,100]$ with at most $5$ digits past the decimal point. Input ends at end of file.

-----Output-----
For each test case, report the $y$-coordinates of the range of locations on the wall that you can hit if you shoot at the mirror. Hitting either side of the (infinitely thin) wall is considered a hit. Report the coordinates from least to greatest. If you can hit an infinitely-long section of the mirror, use the terms negative-infinity or positive-infinity as appropriate. Ignore any potential occlusions, such as the laser hitting you, or the laser hitting the wall before getting to the mirror. Numeric answers should be accurate to within $0.0001$ units. If you can’t hit the wall at all, output can't hit the wall.

-----Examples-----
Sample Input:
5 10 10 10 10 0
5 10 10 5 10 0
6 10 10 10 10 0
Sample Output:
negative-infinity 0
5 12.5
negative-infinity -5
user: Tourists in the numerous Welsh valleys are in need of an IT solution to their transportation troubles. They want to see all the local Points of interest (POIs) in a specific order already set out by their trusty tour guides.

Visitors have to rely on various types of transportation: car, rickshaw, donkey cart, etc. These are only available on call, which is unfortunate as the poor mobile reception in the depths of the idyllic valleys means transportation type can only change at a POI.

Further, a driver will only offer tourists transportation between points $p_ i$, $p_{i+1}$, …, $p_{j-1}$, $p_ j$ under the following conditions:
 - Minimum distance: If the distance is less than a given threshold $d_{\mathrm{min}}$, the itinerary wouldn’t be worth the time of the driver. That is, the total distance of the itinerary: $d_ i + d_{i+1} + \ldots + d_{j-1}$, with $d_ m$ the distance between $p_ m$ and $p_{m+1}$, has to be at least $d_{\mathrm{min}}$.
 - Maximum heading range: Not going straight is perceived as annoying by the cabbies, so the directions traveled can only vary within at most a certain integer amount of $r_{\mathrm{max}}$ degrees.

What the tourists want is a transportation switching scheme, which is a list of increasing indices $s_0$ …$s_{k}$ where points $p_{s_ i}$ are the locations to switch the type of transportation (the same transportation type can be used more than once, but another instance of the same type will need to be hailed once the original driver has had enough).

-----Input-----
 - One line containing the number of modes of transportation $t$ ($1 \le t \leq 200$) followed by the number $n$ ($1 \le n \leq 5 \cdot 10^4$) of points we visit.
 - The next $t$ lines each describe the $i^{\text {th}}$ transportation type with two non-negative integers. The first integer $d_{\mathrm{min}}$ ($0 \leq d_{\mathrm{min}} \le 10^6$) is the minimal distance of each itinerary of this type. The second integer $a$ ($0 \leq a \leq 3.6 \cdot 10^5$) is the maximal heading range in thousandths of a degree.
 - The next $n-1$ lines each contain two integers $d_ i$ and $h_ i$ ($0 \leq d_ i \leq 10^6; -1.8 \cdot 10^5 < h_ i < 1.8 \cdot 10^5$), relative distance and angle from the ${i-1}^{\text {th}}$ point in thousandths of a degree respectively.

-----Output-----
Write one line containing one number $k$: the minimal number of times we have to call for a new type of transportation to visit all $n$ points in the given order, if this is possible. If not, output IMPOSSIBLE.

-----Examples-----
Sample Input:
4 4
100 30000
200 20000
300 10000
400 0
50 10000
75 20000
400 -40000
Sample Output:
2