meetforever/GRPO_CL huggingface dataset:
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 with some other city. Each newly built road can be used in both directions.

Absurdistan is full of absurd coincidences. It took all $N$ cities precisely one year to build their roads. And even more surprisingly, when the roads were finished it was possible to travel from every city to any other city using the newly built roads. We say that such a road network is connected. Being interested in mathematics and probability, you started wondering how unlikely this coincidence really is.

-----Task-----
Each city picked uniformly at random another city to which they built a road. Calculate the probability that the road network ends up being connected.

-----Input-----
The first line contains an integer $N$ $(2\le N\le 140)$ – the number of cities.

-----Output-----
Output one line containing a floating point number denoting the probability that the randomly built road network with $N$ cities and $N$ roads is connected. Your answer should have an absolute error of at most $10^{-8}$.

-----Examples-----
Sample Input:
4
Sample Output:
0.962962962963
user: You are the boss of ACM (Association for Convex Main Office), an upstanding company with a single goal of world domination.

Today, you have decided to move your main office from Helsinki to Singapore. You have purchased a square-shaped land in Singapore, represented as a square on a two dimensional Cartesian plane. The size of your land is $(4 \cdot 10^7) \times (4 \cdot 10^7)$.

After much deliberation, your engineers have devised the optimal shape of your new main office. It will be a grand building, shaped as a convex polygon on the two dimensional grid (a convex polygon is a simple polygon such that for any two of its vertices, the straight line connecting the two vertices is completely inside the polygon). More specifically, your main office is represented by a polygon with at least $3$ vertices such that:
 - Each vertex is located on a lattice point (lattice points are points which coordinates are integers).
 - The coordinates of all vertices are between $0$ and $4 \cdot 10^7$, inclusively.
 - All vertices are located on distinct points.
 - No three vertices are collinear (that is, no three vertices should be on a line).
 - All the vertices form a convex polygon in some order.

It is a known fact that the number of vertices composing your main office is very important. A lucky number would positively influence the chance of success for all of your evil missions. Thus, given $N$, the number of vertices of the main office, can you generate any one possible way to construct your main office? That is, you should output any possible set of $N$ locations of the $N$ vertices of your main office that has the properties described earlier.

-----Input-----
The first line contains a non-negative integer $3 \leq N \leq 400000$, giving the number of vertices your main office should have.

-----Output-----
Output exactly $N$ lines. Each line consists of two integers $0 \le x, y \le 4 \cdot 10^7$, denoting the coordinates of a vertex making up your main office. The coordinates can be given in any order and must adhere to your main office’s restrictions as described in the problem statement.

Warning: Your program will have to write lots of output. Please use fast output routine to minimize the risk of getting a ‘Time Limit Exceeded’ verdict.

-----Examples-----
Sample Input:
3
Sample Output:
0 0
40000000 0
0 40000000
user: Yes, we know $\ldots $ we’ve used Challenge $24$ before for contest problems. In case you’ve never heard of Challenge $24$ (or have a very short memory) the object of the game is to take $4$ given numbers (the base values) and determine if there is a way to produce the value $24$ from them using the four basic arithmetic operations (and parentheses if needed). For example, given the four base values 3 5 5 2, you can produce $24$ in many ways. Two of them are: 5*5-3+2 and (3+5)*(5-2). Recall that multiplication and division have precedence over addition and subtraction, and that equal-precedence operators are evaluated left-to-right.

This is all very familiar to most of you, but what you probably don’t know is that you can grade the quality of the expressions used to produce $24$. In fact, we’re sure you don’t know this since we’ve just made it up. Here’s how it works: A perfect grade for an expression is $0$. Each use of parentheses adds one point to the grade. Furthermore, each inversion (that is, a swap of two adjacent values) of the original ordering of the four base values adds two points. The first expression above has a grade of $4$, since two inversions are used to move the $3$ to the third position. The second expression has a better grade of $2$ since it uses no inversions but two sets of parentheses. As a further example, the initial set of four base values 3 6 2 3 could produce an expression of grade $3$ — namely (3+6+3)*2 — but it also has a perfect grade $0$ expression — namely 3*6+2*3. Needless to say, the lower the grade the “better” the expression.

Two additional rules we’ll use: $1$) you cannot use unary minus in any expression, so you can’t take the base values 3 5 5 2 and produce the expression -3+5*5+2, and $2$) division can only be used if the result is an integer, so you can’t take the base values 2 3 4 9 and produce the expression 2/3*4*9.

Given a sequence of base values, determine the lowest graded expression resulting in the value $24$. And by the way, the initial set of base values 3 5 5 2 has a grade $1$ expression — can you find it?

-----Input-----
Input consists of a single line containing $4$ base values. All base values are between $1$ and $100$, inclusive.

-----Output-----
Display the lowest grade possible using the sequence of base values. If it is not possible to produce $24$, display impossible.

-----Examples-----
Sample Input:
3 5 5 2
Sample Output:
1
user: Ice cream is a difficult topic. You are at the ice cream store and have some trouble deciding how you want your ice cream. How many scoops? What flavours? In what order? The only way to ensure that you are making the right decisions is to approach the problem systematically.

Therefore, you examine each of the $k$ flavours and estimate that the tastiness of a scoop of the $i$th flavour is $t_ i$. However, you are aware that some flavours complement each other, resulting in a tastiness greater than the sum of the individual flavours, whereas others just do not go very well together. Therefore, you have estimated the additional tastiness experienced whenever a scoop of one flavour is directly on top of a scoop of another, and what happens when you put two scoops of the same flavour on top of each other. The additional tastiness experienced whenever flavour $i$ is on top of flavour $j$ is $u_{i,j}$. Of course, you would like to maximize the total tastiness of your ice cream, but there are two problems.

Firstly, your stomach is, regrettably, finite. Therefore, you do not want to order more that $n$ scoops. You may order fewer scoops, if this is better.

Secondly, ice cream isn’t free. Each scoop costs $a$ gold coins, and the cone costs $b$ gold coins (regardless of the number of scoops of ice cream you buy).

You would like to find the maximum possible tastiness per gold coin ratio. The store has an infinite amount of each flavour.

-----Input-----
The first line of input consists of the integers $n$ ($1 \leq n \leq 2 \cdot 10^9$), $k$ ($1 \leq k \leq 100$), $a$ and $b$ ($1 \leq a,b \leq 200$).

The following line consists of $k$ integers $t_ i$ ($-200 \leq t_ i \leq 200$), the tastiness of each of the flavours.

The following $k$ lines each contain $k$ integers. The $j$th number on the $i$th line is the additional tastiness $u_{i,j}$ ($-200 \leq u_{i,j} \leq 200$).

-----Output-----
If it is impossible to get an ice cream with positive tastiness, display $0$.

Otherwise, display the largest possible value of the quotient of the tastiness and the cost of an ice cream.

Your answer will accepted if it is within a relative or absolute error of at most $10^{-6}$ of the correct answer.

-----Examples-----
Sample Input:
20 3 5 5
0 0 0
0 -10 0
30 0 0
0 0 0
Sample Output:
2
user: An integer is considered handsome if every two of its consecutive digits are of different parity. For a given integer $N$, what is its closest handsome number?

Please note: numbers consisting of only one digit are handsome numbers. The distance of two numbers is the absolute value of their difference.

-----Input-----
The first and only line of input contains a positive integer $N$ that consists of at most thousand digits and is not handsome.

-----Output-----
The first and only line of output must contain the required closest handsome number. If two closest numbers exist, output the smaller number first and then the larger one and separate them by a single space.

-----Examples-----
Sample Input:
13
Sample Output:
12 14
user: It’s Halloween, and you’ve organized a bonfire and dance for the neighborhood children. The $n$ children have gathered into a ring to dance around the fire. Each child is wearing one of two fun yet spooky costumes: orange pumpkin, or black bat. Since it’s dark outside, you can only see a few children at a time, as they pass behind the bonfire. The children are not standing evenly so the number of children you can see at each time differs. In particular, numbering the children $0, 1, \ldots , n-1$ clockwise around the dance circle, at any given time you can see child $i$ in the center of your view, as well as $l_ i$ children before child $i$ and $r_ i$ children after child $i$ around the circle (i.e., child $i-l_ i, \ldots , i-1, i, i+1, \ldots , i+r_ i$, where the indices are of course taken modulo $n$).

To help pass the time while the children dance, you wonder to yourself: suppose you only knew, for each child $i$, whether an even or odd number of the $l_ i+r_ i+1$ children centered at child $i$ is wearing the orange pumpkin costume. Would you be able to uniquely reconstruct what costume each child is wearing? Clearly this is possible when $l_ i=r_ i=0$. But what if $l_ i$ and $r_ i$ are not always zero? Maybe there are multiple possible solutions, or none at all? You decide to investigate, later in the evening once you’re back at your computer.

-----Input-----
The first line of the input consists of a single integer $n$, indicating that there are $n$ children in the ring $(1 \leq n \leq 200000)$. The following $n$ lines describe the children you can see at different times. The $i$th line (indexed starting from zero) contains three space-separated non-negative integers $l_ i$, $r_ i$, $x_ i$ ($l_ i+r_ i+1\leq n,0\leq x_ i\leq 1$): you can see $l_ i+r_ i+1$ children when child $i$ is in the center of view ($l_ i$ to the left and $r_ i$ to the right of child $i$). If $x_ i=0$ then an even number of them are wearing the orange pumpkin costume. If $x_ i=1$ then an odd number of them are wearing the orange pumpkin costume.

-----Output-----
Compute the number of ways of assigning a costume to each child, consistent with your observations. Since this number might be large, print the result modulo $10^9 + 7$. (If it’s impossible to find any costume assignment that matches all parity constraints, print 0).

-----Examples-----
Sample Input:
5
1 0 0
1 0 1
3 0 1
3 0 0
3 0 1
Sample Output:
0
user: Given an grid with $N$ rows and $M$ columns, your task is to find a tour such that: 
 - The tour has length at least two
 - All squares in the grid are visited exactly once
 - Let $a_1, \dots , a_{NM}$ be the sequence of visited squares. Then $d(a_ k$, $a_{k+1}) = 2$ or $3$ for every $1 \leq k < NM$. Also, $d(a_{NM}, a_1) = 2$ or $3$.

Here, $d(p, q)$ denotes the Manhattan distance between $p$ and $q$. More precisely, if $p$ is the square on row $r_ p$ and column $c_ p$ and $q$ is the square on row $r_ q$ and column $c_ q$, then we write $p = (r_ p, c_ p), q = (r_ q, c_ q)$ and $d(p, q) = |r_ p - r_ q| + |c_ p - c_ q|$.

-----Input-----
The first and only line of input consists of two integers $N$ and $M$ ($1 \leq N, M \leq 200$).

-----Output-----
If there is no such tour, output -1. Otherwise, output $N \times M$ lines. The $i$th line contains two space-separated integers, denoting the row and column numbers of $a_ i$. If there is more than one such tour, any one will be accepted.

-----Examples-----
Sample Input:
2 3
Sample Output:
1 1
2 2
1 3
2 1
1 2
2 3
user: Dreaming of revolutionizing the tech industry and making the world a better place, Ging has recently moved to Silicon Valley to found his own startup, Katbook.

Katbook allows users to download cat pictures. Katbook is capable of transferring $t$ bits per second.

There are $n$ species of cats, numbered from $1$ to $n$. There is a demand ratio $d_ i$ for the pictures of the $i$-th species. This means, if there is no further restrictions, the $i$-th cat species should have a ‘fair share’ of $\frac{d_ i}{\sum _{j=1}^{n} d_ j}$ fraction of the total bandwidth $t$.

However, because of infrastructure limitations, the bandwidth for the $i$-th species must be between $a_ i$ and $b_ i$.

You are newly hired by Ging as a network engineer at Katbook. Your first task is to find the most ‘fair’ bandwidth allocation satisfying the above constraints. More formally, let $x_ i$ be the bandwidth allocated for downloading pictures of the $i$-th species. You must find $x_ i$ such that:
 - $a_ i \le x_ i \le b_ i$,
 - $\sum _{i=1}^{n} x_ i = t$,
 - Let $y_ i = t \cdot \frac{d_ i}{\sum _{j=1}^{n} d_ j}$ ($y_ i$ is the ‘fair share’ bandwidth, if there was no constraints regarding $a_ i$ and $b_ i$), the value $\sum _{i=1}^{n} \frac{(x_ i - y_ i)^2}{y_ i}$ should be as small as possible.

-----Input-----
The first line of input contains $2$ integers $n$ and $t$ $(1 \le n \le 10^5, 1 \le t \le 10^6)$.

In the next $n$ lines, the $i$-th line contains $3$ integers $a_ i$, $b_ i$ and $d_ i$ $(0 \le a_ i \le b_ i \le 10^6, 0 < b_ i, 1 \le d_ i \le 10^6)$.

It is guaranteed that there exists at least one valid solution.

It can be shown that under these constraints, the solution is unique.

-----Output-----
Output $n$ lines, each line contains the allocation for downloading pictures of the $i$-th species. The answer will be accepted if and only if its relative or absolute error to the optimal solution is at most $10^{-6}$.

Formally, for each line, let your answer be $a$, and the jury’s answer be $b$. Your answer is accepted if and only if $\frac{|a - b|}{\max {(1, |b|)}} \le 10^{-6}$.

-----Explanation of Sample input-----
In the first sample, each cat species has its ‘fair share’ of $\frac{1}{3}$ of the whole bandwidth.

In the second sample, note that the cat species $1$ has much more demand and could get almost all of the available bandwidth if not for its maximum to be capped at $1$. The rest is divided with ratio $2:1$, hence the cat species $2$ gets $6$ bits per second and the cat species $3$ gets $3$ bits per second.

-----Examples-----
Sample Input:
3 10
0 10 1
0 10 1
0 10 1
Sample Output:
3.33333333
3.33333333
3.33333333
user: Julia is betting on a large sporting competition involving matches between pairs of teams. There are no parallel matches and each bettor receives one point for every correct bet they make. Julia had a good streak and is in the lead. Now she worries that her good luck may be turning, and decides to change her strategy.

She collaborates with a betting shop owner who tells her the bets made by everyone else. Whenever Julia makes a bet, she first checks the bets of all bettors with the most points so far (except herself of course) and then chooses the same team as the majority. In the case of a tie, she bets on her favourite of the two teams in the game.

Julia wants to know for how many more matches she is guaranteed to stay in the lead in the worst case (i.e., no matter what bets the others make or what the outcomes of the games are). For this problem we consider Julia to be in the lead if there is no other bettor that has strictly more points than her.

For example, suppose as in Sample Input 1 that Julia initially has three points, and there are two other bettors with three and two points respectively. For the first match, she will make the same bet as the other bettor with three points. If this person bets on the losing team and the other bets on the winning team all players have an equal score of three points. If for the next match the other two persons bet on different teams, Julia places the bet on her favourite, which of course may lose. Thus after two more matches she might lose the lead.

-----Input-----
The input consists of:
 - One line with an integer $n$ ($3 \le n \le 10^5$), the number of people who place their bets.
 - One line with $n$ integers $p_1, \ldots , p_ n$ ($0\le p_ i \le 10^{16}$ for each $i$), the points of all people who play the betting game. The first of these numbers corresponds to the score of Julia. You may assume that no other score exceeds Julia’s score in the beginning.

-----Output-----
Output the number of matches for which Julia is guaranteed to stay in the lead.

-----Examples-----
Sample Input:
3
3 3 2
Sample Output:
1
user: While browsing YouTube videos, you commonly use the handy dandy tool that is Add Video to Watch Later. One day you finally decide that ‘Later’ has finally arrived and you have SO many videos in your list.

You have a list of videos with different types. For example, some might be bouldering videos, some might be cat videos, and so on. You want to watch all videos of the same type before you watch videos of a different type, but you are allowed to watch the video types in any order you wish. For example, you might want to watch all bouldering videos before watching any cat videos.

To start watching, you have to click on a video to play it. You may click on any video in your list to start watching that video. Whenever a video finishes playing, it is automatically deleted from the list. The order of the remaining videos does not change when a video finishes playing. Also, the next video in the list is automatically played if it is of the same type as the video you just watched. If it is of a different type, or if there is no video after the one you just watched, you must click on another video in the list to watch more videos (unless you have watched all videos).

Given the description of your Watch Later list, what is the minimum number of clicks needed to watch every video with the restrictions described above?

-----Input-----
The first line of the input contains two integers $n$ ($1 \leq n \leq 400$), the number of videos in your Watch Later list, and $k$ ($1 \leq k \leq 20$), the number of different video types in the list.

The second line of input contains a string of length $n$ which describes the Watch Later list. The $i^\textrm {th}$ character in the string is a lowercase English letter which describes the type of the $i^\textrm {th}$ video. Two videos are of the same type only if they are denoted by the same letter.

-----Output-----
Output a single integer on a line, indicating the minimum number of clicks required to watch all videos currently in your Watch Later list.

-----Examples-----
Sample Input:
4 2
abba
Sample Output:
2
user: Since you are in Britain, you definitely want to try British food. Unfortunately you will only have a single free evening, so you decided to try all the food you can get in one run. You plan a gigantic meal where you eat one British dish after the other. Clearly not every order of dishes is appropriate. For example, it is not acceptable to eat Blood Pudding directly after Cornish Hevva Cake, but it would be perfectly fine if you chose to eat Baked Beans in between.

You have compiled a comprehensive list of British dishes. For each dish you have also decided which other dishes are fit to be eaten directly afterwards. A menu is a sequence of dishes such that each dish (except the first) is fit to be eaten directly after the previous dish.

After some time studying the list of dishes, you noticed something odd: Whenever it is possible to find a menu in which a dish occurs twice (for example dishes $A$, then $B$, then $C$, then dish $A$ again), there can be at most four different types of dishes between the dish that occurred twice – excluding that dish itself. For example, it is impossible to find a menu like $A, B, C, D, E, F, A$, but it may be possible to find menus like $A, B, C, B, C, B, C, B, C, B, A$ or $A,B,C,D,E,A,B,C,D,E,A$.

But who wants to eat the same dish twice anyway? Clearly, you want to know how many dishes there can be in a menu without repeating any dish!

-----Input-----
The input consists of:
 - One line with two integers $n,m$ ($1 \leq n \leq 10^5$, $1\leq m \leq 10^6$), the number of dishes and compatibilities.
 - $m$ lines, each containing two integers $a$ and $b$ ($1 \leq a,b \leq n$), indicating that you can eat dish $b$ immediately after dish $a$.

Dishes are numbered from $1$ to $n$ in no particular order, and the compatibilities satisfy the constraint described above.

-----Output-----
A single integer indicating the maximum number of courses in a menu without repeated dishes.

-----Examples-----
Sample Input:
4 3
1 2
2 3
2 4
Sample Output:
3
user: Given a registry of all houses in your state or province, you would like to know the minimum size of an axis-aligned square zone such that every house in a range of addresses lies in the zone or on its border. The zoning is a bit lenient and you can ignore any one house from the range to make the zone smaller.

The addresses are given as integers from $1..n$. Zoning requests are given as a consecutive range of houses. A valid zone is the smallest axis-aligned square that contains all of the points in the range, ignoring at most one.

Given the ($x,y$) locations of houses in your state or province, and a list of zoning requests, you must figure out for each request: What is the length of a side of the smallest axis-aligned square zone that contains all of the houses in the zoning request, possibly ignoring one house?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will begin with a line containing two integers $n$ and $q$ ($1 \le n,q \le 10^5$), where $n$ is the number of houses, and $q$ is the number of zoning requests.

The next $n$ lines will each contain two integers, $x$ and $y$ ($-10^9 \le x,y \le 10^9$), which are the ($x$,$y$) coordinates of a house in your state or province. The address of this house corresponds with the order in the input. The first house has address $1$, the second house has address $2$, and so on. No two houses will be at the same location.

The next $q$ lines will contain two integers $a$ and $b$ ($1 \le a < b \le n$), which represents a zoning request for houses with addresses in the range $[a..b]$ inclusive.

-----Output-----
Output $q$ lines. On each line print the answer to one of the zoning requests, in order: the side length of the smallest axis-aligned square that contains all of the points of houses with those addresses, if at most one house can be ignored.

-----Examples-----
Sample Input:
3 2
1 0
0 1
1000 1
1 3
2 3
Sample Output:
1
0