meetforever/GRPO_CL huggingface dataset:
user: Many phones use a nine-pivot unlock pattern. The nine pivots are arranged as a $3 \times 3$ grid, and numbered from $1$ to $9$ as shown in the figure. The unlock pattern is a single stroke that starts at any pivot and visits each pivot exactly once. The stroke goes in a straight line segment between each pair of consecutive pivots. It may pass a pivot multiple times but only the first time counts as a visit. The phone unlocks if the pivots are visited in a predefined secret order. The figure illustrates an unlock pattern of $2 \rightarrow 5 \rightarrow 3 \rightarrow 6 \rightarrow 7 \rightarrow 4 \rightarrow 1 \rightarrow 8 \rightarrow 9$. Due to physical constraints, it is not possible for the stroke to pass but not visit a pivot if it has not been visited before. For instance, a pattern that starts with $1 \rightarrow 7 \rightarrow \dots $ is invalid because it must visit pivot $4$ before visiting pivot $7$. However, starting with $4 \rightarrow 1 \rightarrow 7 \dots $ is fine because pivot $4$ is already visited when the stroke passes it again. 

Consider the directional changes along the unlock pattern. The pattern may take a left turn (‘L’), a right turn (‘R’), go straight (‘S’), or turn around (‘A’) at each pivot except for the first and the last pivot. If we write down the directional changes into one string, we get a string of length seven. This is the turn sequence of the unlock pattern. The turn sequence of the unlock pattern shown in the figure is “LRRRSRL”.

Given a turn sequence, with some of the directional changes replaced by question marks (‘?’), which mean we can take any direction at those pivots, how many different unlock patterns are consistent with this turn sequence?

-----Input-----
The input has a single string of length seven. The string consists of characters ‘L’, ‘R’, ‘S’, ‘A’ and ‘?’ that describe the directional changes at the pivots in the order of their visit.

-----Output-----
Output the number of different unlock patterns that have a turn sequence matching the input.

-----Examples-----
Sample Input:
LRRRSRL
Sample Output:
44
user: Löpkass is out running in the forest again. He loves running in the forest. However, there is a programming competition starting in a couple of hours, and he needs to calculate how long it will take him to get home (so that he can stay in the fresh air as long as possible). You have to help him out.

The forest is modelled as a graph with $n$ nodes. Nodes are numbered $0, 1, ..., n-1$. Löpkass is located in node $0$, and his home is located in node $n-1$. Two nodes $v,w$ are connected if $v+1 = w$. Additionally, two nodes are connected if they are both a multiple of some integer $K$. In this problem, we do not consider $0$ to be a multiple of $K$.

-----Input-----
One line with two integers, the number of nodes $1 \leq n \leq 10^{18}$, and the number $1 \leq K \leq 10^{18}$.

-----Output-----
Print a single integer on a single line: the shortest path from node 0 to node $n-1$.

-----Examples-----
Sample Input:
5 2
Sample Output:
3
user: Anthony and Cora are playing Dominion, their favorite card game. In Dominion, there are $T$ different card types, and each player has a set of cards (known as a deck). A deck $D$ is said to have $C$ combos if $C$ is the largest integer such that for $C$ different card types in the game, $D$ contains at least two cards of that type. Anthony currently has $N$ cards and he wants to trade cards with Cora such that he’ll have a deck with exactly $K$ combos.

For each card type $i$ ($1\leq i\leq T$), Anthony can choose to perform at most one transaction. There are two types of transaction:
 - Buy up to two cards of $i^{th}$ type from Cora at $a_ i$ coins each
 - Sell all his cards of $i^{th}$ type for $b_ i$ coins each

Anthony wants to maximize his profit while obtaining a complete deck. Anthony is willing to spend coins in order to obtain a complete deck if necessary, but of course he wants to minimize his spending in that case. Note that he doesn’t care about keeping the rest of his cards which don’t contribute to the complete deck.

Anthony has hired you to help him calculate how much money he can make if he chooses the optimal strategy for obtaining enough combos. If he has to spend money, output a negative number.

-----Input-----
The first line of the input contains three integers $N$, $T$, and $K$, $1\leq K\leq T\leq 100000$, $1\leq N\leq 2T$.

The next line is a list of $N$ integers representing the cards in Anthony’s deck. Each integer on this line is between $1$ and $T$ inclusive. It is guaranteed no integers appear more than twice.

Finally, each of the next $T$ lines of the input contains two integers each. The $i^\mathrm {th}$ line contains $a_ i$ and $b_ i$, $1\leq a_ i, b_ i\leq 10^9$, corresponding to the price of buying and selling a card of type $i$.

-----Output-----
Output a single integer denoting Anthony’s profit assuming he trades optimally.

-----Explanation of Sample Input-----
In the first example, Anthony should sell two of card $1$ and buy one of card $2$ and one of card $3$ for a net profit of $10$ coins. If he chooses to sell one of card $3$ and buy one of card $2$, then he’ll end up spending $20$ coins.

-----Examples-----
Sample Input:
4 3 2
1 3 2 1 
1 50
50 20
40 30
Sample Output:
10
user: Dee Siduous is a botanist who specializes in trees. A lot of her research has to do with the formation of tree rings, and what they say about the growing conditions over the tree’s lifetime. She has a certain theory and wants to run some simulations to see if it holds up to the evidence gathered in the field.

One thing that needs to be done is to determine the expected number of rings given the outline of a tree. Dee has decided to model a cross section of a tree on a two dimenional grid, with the interior of the tree represented by a closed polygon of grid squares. Given this set of squares, she assigns rings from the outer parts of the tree to the inner as follows: calling the non-tree grid squares “ring $0$”, each ring $n$ is made up of all those grid squares that have at least one ring $(n-1)$ square as a neighbor (where neighboring squares are those that share an edge).

An example of this is shown in the figure below.

 Figure D.1 

Most of Dee’s models have been drawn on graph paper, and she has come to you to write a program to do this automatically for her. This way she’ll use less paper and save some $\ldots $ well, you know.

-----Input-----
The input will start with a line containing two positive integers $n$ $m$ specifying the number of rows and columns in the tree grid, where $n, m \leq 100$. After this will be $n$ rows containing $m$ characters each. These characters will be either ‘T’ indicating a tree grid square, or ‘.’.



-----Output-----
Output a grid with the ring numbers. If the number of rings is less than 10, use two characters for each grid square; otherwise use three characters for each grid square. Right justify all ring numbers in the grid squares, and use ‘.’ to fill in the remaining characters.

If a row or column does not contain a ring number it should still be output, filled entirely with ‘.’s.

-----Examples-----
Sample Input:
6 6
.TT...
TTTT..
TTTTT.
TTTTT.
TTTTTT
..T...
Sample Output:
...1.1......
.1.2.2.1....
.1.2.3.2.1..
.1.2.3.2.1..
.1.1.2.1.1.1
.....1......
user: You might be familiar with the binomial coefficient ${m \choose k}$ defined as ${m \choose k} = \frac{m!}{k!(m-k)!}$, where $m$ and $k$ are non-negative integers and $k \leq m$. Let $T_2(n)$ be the number of odd binomial coefficients such that $0 \le k \le m < n$. The most useful mathematical inequality you will learn during this competition is\[ 0.812556 n^{\log _2 3} \le T_2(n) \le n^{\log _2 3}. \]

Emma doesn’t like such imprecise inequalities and would like to calculate $T_2(n)$ exactly. Can you help her?

-----Input-----
The input contains one line with one integer $n, 1 \leq n \le 10^{11}$.

-----Output-----
Output one line with the value of $T_2(n)$.

-----Examples-----
Sample Input:
4
Sample Output:
9
user: Oliver is a manager of a bank near KTH and wants to close soon. There are many people standing in the queue wanting to put cash into their accounts after they heard that the bank increased the interest rates by $42\% $ (from $0.01\% $ per year to $0.0142\% $ per year). 

However, there are too many people and only one counter is open which can serve one person per minute. Greedy as Oliver is, he would like to select some people in the queue, so that the total amount of cash stored by these people is as big as possible and that money then can work for the bank overnight.

There is a problem, though. Some people don’t have the time to wait until the bank closes because they have to run somewhere else, so they have to be served before a certain time, after which they just leave. Oliver also turned off the infrared door sensor outside the bank, so that no more people can enter, because it’s already too crowded in the hall.

-----Task-----
Help Oliver calculate how much cash he can get from the people currently standing in the queue before the bank closes by serving at most one person per minute.

-----Input-----
The first line of input contains two integers $N$ $(1\leq N\leq 10000)$ and $T$ $(1\le T \le 47)$, the number of people in the queue and the time in minutes until Oliver closes the bank. Then follow $N$ lines, each with 2 integers $c_ i$ and $t_ i$, denoting the amount of cash in Swedish crowns person $i$ has and the time in minutes from now after which person $i$ leaves if not served. Note that it takes one minute to serve a person and you must begin serving a person at time $t_ i$ at the latest. You can assume that $1\le c_ i \le 100000$ and $0 \le t_ i < T$.

-----Output-----
Output one line with the maximum amount of money you can get from the people in the queue before the bank closes.

-----Examples-----
Sample Input:
4 4
1000 1
2000 2
500 2
1200 0
Sample Output:
4200
user: Neelix has been working on making the Altonian warp bubble power itself up. While waiting for the system checks to complete, he saw a number on his screen and wondered if there is a number that powers itself up to this number.

-----Input-----
The first and only input line consists of a single integer $n$, the number Neelix saw on his screen.

-----Output-----
Output the number $x$, where $x^x = n$. Your answer must have an absolute or relative error of at most $10^{-6}$.

-----Limits-----
 - $1 \leq n \leq 10000000000$

-----Examples-----
Sample Input:
4
Sample Output:
2.0
user: Robert Hood, a less famous sibling of the Robin Hood, is fed up. Despite him being a young, talented archer he never seems to reach quite the same level as his legendary brother, and so he decided to come up with rules for a new archery contest, in which he will stand a better chance of winning.

The rules for the new kind of archery contest are quite simple: the winner is no longer the one who can score the most points, but instead the one who can achieve the longest distance between any pair of arrows hitting the target. Your task is to write the code to calculate that distance.

Each contestant is allowed a number of arrow shots, and the coordinates of the arrows successfully hitting the target are given as a list of pairs. The coordinate system is Cartesian with the origin in the centre of the archery butt. If a contestant does not hit the target with at least two arrows he or she is disqualified and removed from the input data.

-----Input-----
The input starts with a line containing a single positive integer $C, 2 \le C \le 100000$, representing the number of shots for this particular contestant. Each following line contains a pair of integer coordinates separated by a space, representing the $x$- and $y$-coordinates of a successful shot. The absolute value of any coordinate does not exceed $1000$.

-----Output-----
Print the longest distance between any pair of arrows as a floating point number on a single line. The answer is considered correct if it has a relative or absolute error of less than $10^{-6}$.

-----Examples-----
Sample Input:
2
2 2
-1 -2
Sample Output:
5.0
user: Pumpkin Pete is trying out a new type of “rapid-growth” pumpkin seed that he bought from the farmer’s market. Without looking at the directions, Pumpkin Pete tears through the packaging and plants the seeds he has into his pumpkin patch. Unbeknownst to Pumpkin Pete, his rival, Gourd Gary, is watching him plant the new seeds from a secret vantage point. After Pumpkin Pete leaves the pumpkin patch, Gourd Gary approaches the patch and picks up the packaging that Pumpkin Pete left on the ground. The packaging says the following:
 - A pumpkin starts with four roots of length zero.
 - Each of the pumpkin’s four roots grow a single unit in a different cardinal direction each day.
 - When a pumpkin dies, its remains will not disappear.
 - If any of the roots grow into another pumpkin or its roots – dead or alive – the pumpkin will die at the end of that day.
 - Roots cannot grow outside of the bounds of a plot. In other words, a pumpkin will die if one of its roots tries to go outside the bounds of the pumpkin patch.
 - If the roots of multiple pumpkins reach the same spot on the same day, each one of the affected roots stops growing (i.e. fight for nutrients) and in turn, the pumpkins will die at the end of the day.
 - When a pumpkin dies, its roots do not grow on subsequent days.

With this information and the knowledge of where each of the pumpkin seeds were planted, Gourd Gary starts to think about which pumpkins would still be alive if they were left to grow for $D$ days$\ldots $Input

The first line of input contains three integers: the number of pumpkins $P$, the number of days $D$ that will pass ($1 \leq D \leq 10$), and $N$ ($1 \leq N \leq 30, 1 \leq P \leq N^2$) the dimension of the $N\times N$ grid. The next $P$ lines of input contain two integers, $R$ and $C$ ($0 \leq R,C < N$), representing the row and column position of each pumpkin. No two pumpkins will be at the same position. Position $(0,0)$ is the top left corner of the grid.

-----Output-----
The output will consist of a single line per pumpkin in the same relative order as the input. If the pumpkin is alive after $D$ days have passed, print “ALIVE”. Otherwise, print the day the pumpkin died as a single integer.

-----Examples-----
Sample Input:
4 2 8
3 2
5 5
4 3
1 1
Sample Output:
1
2
1
2
user: Quido and Hugo are making a chocolate cake. The central ingredient of the cake is a large chocolate bar, lying unwrapped on the kitchen table. The bar is an $M \times N$ rectangular grid of chocolate blocks. All of the $MN$ blocks are rectangles of identical shape and size. The chocolate bar is of top quality and the friends want to eat part of it, before the rest is used in the cake.

“OK,” says Quido, “let’s divide the whole bar into two triangular chunks by a straight diagonal cut from its upper-left corner to its lower-right corner. We will then eat all of the blocks which have been cut exactly in half, into two equal-area pieces. You will eat one half and I will eat the other half of each such block. All other blocks, that is, the blocks which are either uncut or cut into two parts of different sizes, will go directly into the cake. Of course, we will make sure the cut is perfectly precise.

Let’s see how much chocolate we get to eat!”

-----Input-----
The input consists of two space-separated integers $M$ and $N$ given on a single line, (where $1 \leq M, N \leq 10^{18}$). The numbers $M$ and $N$ denote the number of blocks in one column and in one row, respectively, in the chocolate bar.

-----Output-----
Print the number of blocks of the chocolate bar which are cut into exactly two pieces of equal area.

-----Examples-----
Sample Input:
6 10
Sample Output:
2
user: Penelope is part of the admin team of the newly built supercomputer. Her job is to assign workstations to the researchers who come here to run their computations at the supercomputer. 

Penelope is very lazy and hates unlocking machines for the arriving researchers. She can unlock the machines remotely from her desk, but does not feel that this menial task matches her qualifications. Should she decide to ignore the security guidelines she could simply ask the researchers not to lock their workstations when they leave, and then assign new researchers to workstations that are not used any more but that are still unlocked. That way, she only needs to unlock each workstation for the first researcher using it, which would be a huge improvement for Penelope.

Unfortunately, unused workstations lock themselves automatically if they are unused for more than $m$ minutes. After a workstation has locked itself, Penelope has to unlock it again for the next researcher using it. Given the exact schedule of arriving and leaving researchers, can you tell Penelope how many unlockings she may save by asking the researchers not to lock their workstations when they leave and assigning arriving researchers to workstations in an optimal way? You may assume that there are always enough workstations available.

-----Input-----
The input consists of:
 - one line with two integers $n$ ($1 \leq n \leq 300000$), the number of researchers, and $m$ ($1 \leq m \leq 10^8$), the number of minutes of inactivity after which a workstation locks itself;
 - $n$ lines each with two integers $a$ and $s$ ($1 \leq a, s \leq 10^8$), representing a researcher that arrives after $a$ minutes and stays for exactly $s$ minutes.

-----Output-----
Output the maximum number of unlockings Penelope may save herself.

-----Examples-----
Sample Input:
3 5
1 5
6 3
14 6
Sample Output:
2
user: You are leading a kayaking trip with a mixed group of participants in the Stockholm archipelago, but as you are about to begin your final stretch back to the mainland you notice a storm on the horizon. You had better paddle as fast as you can to make sure you do not get trapped on one of the islands. Of course, you cannot leave anyone behind, so your speed will be determined by the slowest kayak. Time to start thinking; How should you distribute the participants among the kayaks to maximize your chance of reaching the mainland safely? 

The kayaks are of different types and have different amounts of packing, so some are more easily paddled than others. This is captured by a speed factor $c$ that you have already figured out for each kayak. The final speed $v$ of a kayak, however, is also determined by the strengths $s_1$ and $s_2$ of the two people in the kayak, by the relation $v=c(s_1+s_2)$. In your group you have some beginners with a kayaking strength of $s_ b$, a number of normal participants with strength $s_ n$ and some quite experienced strong kayakers with strength $s_ e$.

-----Input-----
The first line of input contains three non-negative integers $b$, $n$, and $e$, denoting the number of beginners, normal participants, and experienced kayakers, respectively. The total number of participants, $b+n+e$, will be even, at least $2$, and no more than $100000$. This is followed by a line with three integers $s_ b$, $s_ n$, and $s_ e$, giving the strengths of the corresponding participants ($1 \leq s_ b < s_ n < s_ e \leq 1000$). The third and final line contains $m = \frac{b+n+e}{2}$ integers $c_1, \ldots , c_ m$ ($1 \leq c_ i \leq 100000$ for each $i$), each giving the speed factor of one kayak.

-----Output-----
Output a single integer, the maximum speed that the slowest kayak can get by distributing the participants two in each kayak.

-----Examples-----
Sample Input:
3 1 0
40 60 90
18 20
Sample Output:
1600
user: In the card game Wolf (a variant of the famous card games War and Svälta räv), two players both have a pile of cards, which together form a normal deck of playing cards. Such a deck contains $52$ cards, with each card being a unique combination of one of $4$ suits and $13$ ranks.

During a turn, each player picks the topmost card from their piles, and shows it. This process is repeated until the two cards have the same suit. Then, the player whose card had the highest rank receives all the cards that have been shown by the players during the turn (including the two that had the same suit), and these cards are shuffled into that player’s pile. The turn then ends. The turn also ends if any player, when they are supposed to pick a card from their pile, has an empty pile. A player wins if, after the end of a turn, she has cards remaining in her pile but the opposing player has no cards remaining in their pile. This means that the game is tied if neither player has cards in their pile after the end of a turn.

Your opponent is currently taking a bathroom break, so you have the opportunity to reshuffle your piles. You cannot exchange any cards, since your opponent knows the set of cards in their pile. Is it possible to reshuffle each of the two piles, so that you have won the game after the next turn?

-----Input-----
The first line of input contains the number of cards in your hand ($0 \leq n \leq 52$). Each of the next $n$ lines contains a card specified by a number from $1$ to $13$ and one of the letters ’C’, ’D’, ’H’, ’S’, separated by a space.

The letter of a card specifies its suit (either clubs, diamonds, hearts or spades, respectively). The number of a card specifies the rank of the card. For simplicity, aces are represented as a $1$, and jacks, queens and kings are represented as the numbers $11$, $12$ and $13$. For the purpose of this problem, we consider a rank $a$ to be lower than a rank $b$ if the number of rank $a$ is lower than the number of rank $b$ (meaning aces are of the lowest rank).

-----Output-----
Output “possible” if it is possible to reshuffle the decks, and “impossible” if it is not.

-----Examples-----
Sample Input:
28
1 C
2 C
3 C
4 C
5 C
6 C
7 C
1 D
2 D
3 D
4 D
5 D
6 D
7 D
1 H
2 H
3 H
4 H
5 H
6 H
7 H
1 S
2 S
3 S
4 S
5 S
6 S
7 S
Sample Output:
possible
user: A new student dorm has been opened! It consists of $M$ buildings, labeled with integers from $1$ to $M$. The dorm is initially empty, but soon $N$ students will be moving in at a rate of exactly one student per day.

Each time a new student moves in a building, a big party is being held inside that building. The noise of the party is equal to the number of students located inside the building. The dorm management is not particularly fond of noise, so they will occasionally empty a certain building to keep the parties at a reasonable noise level. They do this by moving all its residents to a completely different student dorm. The management can decide to do this after any day, but they realized that it doesn’t pay off to do it more than $K$ times.

Help the management! Knowing which buildings are being moved in by students, determine the minimal possible total noise level (the sum of noise levels of all $N$ parties) that can be achieved with emptying some of the buildings at most $K$ times.

-----Input-----
The first line of input contains the integers $N$ ($1 \leq N \leq 1000000$), $M$ ($1 \leq M \leq 100$) and $K$ ($1 \leq K \leq 500$) from the task description.

The $i$-th line, out of $N$ in total, contains an integer from the interval $[1,M]$: the label of the building where a student is moving in on the $i$-th day.

-----Output-----
The first and only line of output must contain the required minimal possible total noise level.

-----Examples-----
Sample Input:
5 1 2
1
1
1
1
1
Sample Output:
7