meetforever/GRPO_CL huggingface dataset:
user: Magnus is the youngest chess grandmaster ever. He loves chess so much that he decided to decorate his home with chess pieces. To decorate his long corridor, he decided to use the knight pieces. His corridor is covered by beautiful square marble tiles of alternating colors, just like a chess board, with $n$ rows and $m$ columns. He will put images of knights on some (possibly none) of these tiles. Each tile will contain at most one knight.

The special thing about his arrangement is that there won’t be any pair of knights can attack each other. Two knights can attack each other if they are placed in two opposite corner cells of a 2 by 3 rectangle. In this diagram, the knight can attack any of the Xs.

Given the dimension of the long corridor, your task is to calculate how many ways Magnus can arrange his knights. Two arrangements are considered different if there exists a tile which contains a knight in one arrangement but not in the other arrangement (in other words, rotations and reflections are considered different arrangements).

-----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 consist of a single line with two integers $n$ and $m$ ($1 \le n \le 4$, $1 \le m \le 10^{9}$) representing the dimensions of the carpet. There will be a single space between $n$ and $m$.

-----Output-----
Output a single line with a single integer representing the number of possible arrangements, modulo $(10^{9}+9)$.

-----Examples-----
Sample Input 1:
1 2
Sample Output 1:
4

Sample Input 2:
2 2
Sample Output 2:
16
user: Mirko’s ASCII street is made of $N$ lowercase letters of the English alphabet. The city government occasionally replaces the tiles in the street. However, the letter tiles are in high demand, so the government has only $M$ different tile patterns available.

The $i$th tile pattern consists of $L_ i$ letters. A tile cannot be rotated or broken into pieces, and it can only be placed such that the tile letters coincide with the contiguous letter subsequence in the street. Tiles can overlap and we can use multiple tiles of the same pattern.

A street cell is untileable if it cannot be covered by any tile. Compute the number of untileable cells.

-----Input-----
The first line of input contains the positive integer $N$ ($1 \le N \le 300000$), the length of the street.

The second line of input contains $N$ lowercase English letters, the letter sequence in the street.

The third line of input contains the positive integer $M$ ($1 \le M \le 5000$), the number of tile patterns.

Each of the next $M$ lines contains a description of a tile pattern with length $L_ i$ ($1 \le L_ i \le 5000$). The tile pattern descriptions consist of lowercase English letters.

The total length of the tile patterns (the sum of the $L_ i$’s) is at most $13000000$.

-----Output-----
The first and only line of output must contain the required number of untileable cells.

-----Examples-----
Sample Input 1:
6
abcbab
2
cb
cbab
Sample Output 1:
2

Sample Input 2:
4
abab
2
bac
baba
Sample Output 2:
4
user: An oft-forgotten part of a well-rounded software engineer’s training is those long but vital months spent learning the art of shipwrighting.

Modern boats, as we know, are superbly safe, to the point that they are nigh unsinkable. Even in a head-on collision the ship can be saved by a system of bulkheads, reinforced vertical sections inside the structure designed to prevent ingressed water from spreading to other sections.

A splendid new ship of the line has had a team of talented artists hard at work reticulating the finest splines for use in this vessel. However, not being concerned with such details, they have left out the placements of the bulkheads as an exercise for their readers.

This is where you can help. First, we need to find how many bulkheads we can fit in the ship. Second, the exact placements of the bulkheads need to be found.

-----Input-----
 - One line containing an integer $C$ ($10 \le C \le 10^9$), the minimum area of a bulkhead section.
 - One line containing an integer $N$ ($3 \le N \le 10^5$), the number of vertices in the artists’ design for the boat.
 - $N$ unique lines, each containing two integers: $x$ and $y$ ($-10^5 \le x, y \le 10^5$), the coordinates of the vertices from the hull in counter-clockwise winding order.

The shape of the boat never doubles back on itself horizontally; that is to say, if a vertical line is drawn through the cross-section, no matter where, it will always pass through the boat exactly once—never twice—however it may run along one or more edges.

It is guaranteed that it is always possible to fit at least one bulkhead section into the ship.

-----Output-----
 - One line containing one integer, $M$: the maximum number of bulkhead sections that can be created. It is guaranteed that $M$ is between $1$ and $100$.
 - $M-1$ lines, each containing one real number: the $X$-coordinate of the placement of a bulkhead such that the sections defined by it have equal area to all the others. Bulkhead placements must be given in increasing order of $X$.

All output must be accurate to an absolute or relative error of at most $10^{-6}$.

-----Examples-----
Sample Input 1:
50
4
110 10
80 10
80 0
110 0
Sample Output 1:
6
85
90
95
100
105

Sample Input 2:
24
3
10 10
30 10
20 20
Sample Output 2:
4
17.071067
20
22.928932
user: Mirko got a set of intervals for his birthday. There are many games he can play with them. In one of them, Mirko must find the longest sequence of distinct intervals such that each interval in the sequence is in the set and that each interval contains the one that follows in the sequence.

Write a program which finds one such longest sequence.

-----Input-----
The first line of input contains the integer $N$ $(1 \le N \le 100000)$, the number of intervals in the set.

Each of the following $N$ lines contains two integers $A$ and $B$ describing one interval $(1 \le A < B \le 1000000)$.

-----Output-----
Output the length $K$ of the longest sequence on the first line.

Each of the following $K$ lines should contain one element of the sequence, an interval in the same format it was given in the input.

-----Examples-----
Sample Input 1:
3
3 4
2 5
1 6
Sample Output 1:
3
1 6
2 5
3 4

Sample Input 2:
5
10 30
20 40
30 50
10 60
30 40
Sample Output 2:
3
10 60
30 50
30 40
user: Your greeting card company makes unique greeting cards. The sizes of these greeting cards vary widely because of the whims of card designers. There are a lot of different types of cards, and each has a specific quantity that you need to manufacture.

Your job is to determine what envelopes to order for these greeting cards. You have a strict limit on the different number of different sizes of envelopes, which may be less than the number of distinct sizes of cards. You need to have envelopes so that every card fits in some envelope, possibly with room to spare, and the amount of waste paper is minimized. Measure the waste paper by the area of the envelope that is in excess of the area of the card, for each card. For example, a $10 \times 4$ card in a $10 \times 4$ envelope has no wasted paper, but a $10 \times 4$ card in a $12 \times 5$ envelope has waste of $20$. You may not rotate the cards to fit them in the envelopes better.

Suppose that you have $5$ types of cards: $10 \times 10$ ($5$ of these), $9 \times 8$ ($10$ of these), $4 \times 12$ ($20$ of these), $12 \times 4$ ($8$ of these), and $2 \times 3$ ($16$ of these).

Now, suppose that you can only buy one type of envelope. Since all cards have to fit in that one envelope size, the smallest envelope size you can use is $12 \times 12$, with an area of $144$. The wastes by each type of card are $144 - 10 \cdot 10=44$, $144 - 9 \cdot 8=72$, $144 - 4 \cdot 12=96$, $144 - 12 \cdot 4=96$, and $144 - 2 \cdot 3=138$, respectively. The total waste is $44 \cdot 5 + 72 \cdot 10 + 96 \cdot 20 + 96 \cdot 8 + 138 \cdot 16=5836$.

Suppose that you can buy $2$ types of envelopes. The best you can do is to put the $10 \times 10$, $9 \times 8$ and $12 \times 4$ cards in $12 \times 10$ envelopes, and the $4 \times 12$ and $2 \times 3$ cards in $4 \times 12$ envelopes. That adds up to waste of $1828$.

If you can buy $5$ types of envelopes, then you can match one envelope type to each card type, and there’s no waste!

Given a list of card types and the number of types of envelopes you can buy, what is the smallest amount of wasted paper you can achieve?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of the input will consist of two space-separated integers $n$ and $k$ ($1 \le n,k \le 15$), where $n$ is the number of different types of cards, and $k$ is the maximum number of types of envelopes you can order. Each of the following $n$ lines will consist of three integers, describing a type of card. The integers are $w$, $h$ and $q$ ($1 \le w,h,q \le 10000$), where $w$ is the width of the cards of this type, $h$ is the height of the cards, and $q$ is the quantity of cards of this type.

-----Output-----
Output a single integer, representing the smallest possible total amount of wasted paper.

-----Examples-----
Sample Input 1:
5 1
10 10 5
9 8 10
4 12 20
12 4 8
2 3 16
Sample Output 1:
5836

Sample Input 2:
5 2
10 10 5
9 8 10
4 12 20
12 4 8
2 3 16
Sample Output 2:
1828
user: In a few weeks time, a new president will be elected in the country of Marecia. There are two political parties: the Constituents and the Federals. Each party has one presidential candidate; one of these will become the new president.

This year you are more involved than ever before: you are the candidate for the Constituents! With only a few weeks to go, you decide to take stock of the situation. Instead of touring blindly through the country, you feel that it is better to focus your attention on certain particular states.

The Marecian electoral system is quite complicated. Every registered voter will vote for exactly one of the two candidates, but the totals are not calculated nationwide. Instead, every state has a fixed number of delegates. After the popular vote is in, the delegates from all states travel by foot to the capital of Marecia, the city of Ashwington. One final round of voting is done in Ashwington: every delegate gets to vote for one of the two candidates. A delegate is required by law to vote for the candidate who received the majority of the votes in his own home state.

Your campaign team has painstakingly compiled a very accurate list of voting preferences per state. Therefore you now have the following information about every state:
 - The number of voters who will definitely vote for the Constituents;
 - The number of voters who will certainly vote for the Federals;
 - The number of voters who have not made up their minds yet.

Voters from the first two categories are not susceptible to any further campaigning, so your focus for the remaining weeks is on people of the third category. Specifically, you would like to know the minimum number of people you still have to convince in order to secure a victory.

If there is a tie on either state or national level, the law states that it is resolved in favour of the oldest of the two political parties, which unfortunately is the Federal party.

-----Input-----
The first line contains a single integer $S$, the number of states.

The following $S$ lines each contain four integers $D_ i$, $C_ i$, $F_ i$, $U_ i$, where
 - $D_ i$ denotes the number of delegates for the $i$-th state,
 - $C_ i$ denotes the number of registered voters in the $i$-th state who will definitely vote for the Constituents,
 - $F_ i$ denotes the number of registered voters in the $i$-th state who will certainly vote for the Federals,
 - $U_ i$ denotes the number of undecided voters in the $i$-th state.

There are at most $2016$ delegates in total, and the total number of registered voters is at most $10^9$. Every state has at least one delegate and at least one registered voter. There is at least one state.

-----Output-----
Output one line with a single integer: the minimum number of voters you have to convince to secure a victory. If it is not possible for you to win the election, output “impossible” instead.

-----Examples-----
Sample Input 1:
3
7 2401 3299 0
6 2401 2399 0
2 750 750 99
Sample Output 1:
50

Sample Input 2:
3
7 100 200 200
8 100 300 200
9 100 400 200
Sample Output 2:
impossible
user: Slavko decided to challenge Mirko! He gave him a real number $P$ and a bag full of pieces of paper with exactly one number between $1$ and $5$ inclusive written on each paper. There is an unlimited quantity of each type of paper.

Mirko’s task is to pick the minimum number of papers in a way that the average of the numbers written on them equals exactly $P$.

-----Input-----
First and only line of input contains real number $P$. $P$ will have between 1 and 9 decimal places, inclusive $(1 \leq P \leq 5)$.

-----Output-----
First and only line of output should contain five nonnegative integers – numbers of ones, twos, threes, fours and fives used, respectively. If there are multiple solutions, output any one of them.

-----Examples-----
Sample Input 1:
5.0
Sample Output 1:
0 0 0 0 1

Sample Input 2:
4.5
Sample Output 2:
0 0 0 1 1
user: A hopper is a virtual creature that visits Java programs and explores their arrays. Scientists observed a hopper and came to the following conclusions:
 - a hopper only visits arrays with integer entries,
 - a hopper always explores a sequence of array elements using the following rules:
 - a hopper cannot jump too far, that is, the next element is always at most $D$ indices away (how far a hopper can jump depends on the length of its legs),
 - a hopper doesn’t like big changes in values—the next element differs from the current element by at most $M$, more precisely the absolute value of the difference is at most $M$ (how big a change in values a hopper can handle depends on the length of its arms), and
 - a hopper never visits the same element twice.
 - a hopper will explore the array with the longest exploration sequence.

The scientists now need to prepare arrays for further study of hoppers and they need your help. They want a program that given an array and values $D$ and $M$ computes the length of the longest exploration sequence in the array.

-----Input-----
The first line contains three numbers $n$, $D$, $M$, where $n$ is the length of the array (as described above, $D$ is the maximum length of a jump the hopper can make, and $M$ is the maximum difference in values a hopper can handle). The next line contains $n$ integers—the entries of the array. We have $1 \leq D \leq 7$, $1 \leq M \leq 10000$, $1 \leq n \leq 1000$ and the integers in the array are between $-1000000$ and $1000000$.

-----Output-----
The output contains one line—the length of the longest exploration sequence. The hopper can start and end at any location and the length is computed as the number of visited locations.

-----Examples-----
Sample Input 1:
8 3 1
1 7 8 2 6 4 3 5
Sample Output 1:
8

Sample Input 2:
8 2 1
1 7 8 2 6 4 3 5
Sample Output 2:
3
user: Barr the Bear is playing the game Bits with Swanky Shen!

Bits is a very simple game. At the start, a circle of $N$ non-negative integers $A_1$, $A_2$, $A_3$, $\ldots $, $A_ N$ is shown to both players. That is, to the left of the integer $A_ i$ is the integer $A_{i-1}$ if $i > 1$; and the integer $A_ N$ otherwise. To the right of the integer $A_ i$ is the integer $A_{i+1}$ if $i < N$; and the integer $A_1$ otherwise. Also an integer $K$ is given to both players.

To win this game, one must divide the circle of integers into exactly $K$ contiguous non-empty sections, such that the bitwise AND of the powers of all sections is maximized. The power of a contiguous section of integers is the bitwise OR of all integers in that section.

Barr the Bear is lazy and knows that you are wise with bits. Hence, he has hired you to help him to win the game!

Note: The binary bitwise operators OR and AND operate on the base-$2$ representation of the integers and correspond to the operators | and & respectively in C++ or Java.

-----Input-----
The first line contains integers $N$ and $K$ ($1 \leq K \leq N \leq 5\cdot 10^5$), namely the number of integers and the number of contiguous non-empty sections required.

The next line contains $N$ integers, the $i^\textrm {th}$ of which is the integer $A_ i$ ($0 \leq A_ i \leq 10^9$).

-----Output-----
Output a single integer in one line: The maximum bitwise AND of the powers of the sections in an optimal division of the circle of integers.

-----Explanation-----
In the first sample, the circle is $(2, 3, 4, 1)$. A possible division is $(3, 4)$ and $(1, 2)$. $(3, 4)$ has power $7$ and $(1, 2)$ has power $3$. The bitwise AND of $7$ and $3$ is $3$. Note that a section can possibly wrap around the circle.

In the second sample, a possible division is $(2, 2, 4)$, $(4)$, $(4, 2)$. The sections’ powers are $6$, $4$ and $6$ respectively, which have bitwise AND of $4$. Note that we require the sections to be contiguous integers, so the division $(2, 4)$, $(2, 4)$, $(2, 4)$ is not permissible.

In the third sample, we can only have one section. This section will have all the integers, and thus have power $3$.

-----Examples-----
Sample Input 1:
4 2
2 3 4 1
Sample Output 1:
3

Sample Input 2:
6 3
2 2 2 4 4 4
Sample Output 2:
4
user: The Algebraist Carpet Manufacturing (ACM) group likes to produce area carpets based upon various geometric figures. The 2014 ACM carpets are all equilateral triangles. Unfortunately, due to a manufacturing defect, some of the carpets are not as stain-resistant as intended. The ACM group is offering to replace each defective carpet that contains a stain.

The web form used to report the stained carpet requests the three distances that the stain is away from the corners of the rug. Based upon these three numbers, you need to compute the area of the rug that is to be sent to the customer, or indicate that the customer’s carpet doesn’t come from ACM.

-----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 consist of a single line with three floating point numbers $a$, $b$ and $c$ ($0 < a,b,c \le 100$, at most $10$ digits after the decimal point) representing the distances from the stain to each of the three corners of the carpet. There will be a single space between $a$ and $b$, and between $b$ and $c$.

-----Output-----
Output a single line with a single floating point number. If there is a carpet that satisfies the constraints, output the area of this carpet. If not, output -1. The output should have an absolute error of at most $10^{-3}$.

-----Examples-----
Sample Input 1:
1 1 1.732050
Sample Output 1:
1.732050808

Sample Input 2:
1 1 3.0
Sample Output 2:
-1
user: Everyone knows that cats love to eat mice. Naturally, when given the opportunity, any cat wants to eat as many mice as possible.

It just so happens that Cartesian Cat lives on the Cartesian Plane with her home located at $(0, 0)$. Obviously, none of the mice live at this location, they are smarter than that! However, the mice aren’t terribly smart. They live on the plane with Cartesian Cat and at time $t = 0$, stick their heads above the ground, in sight of Cartesian Cat. Each of the mice stays in sight at its location for a specified amount of time before ducking back underground, where Cartesian Cat can’t get it.

Cartesian Cat has a plan. At time $t = 0$, she’ll run straight towards a mouse, eat it instantaneously, and then head for another mouse, repeating the sequence until all the mice are eaten. She encounters two problems however: each time she eats a mouse her velocity reduces (due to her increased mass) by a constant multiplicative factor, $m$. That is, if her velocity is $v$ before eating a mouse, then after eating the mouse her velocity is $v\cdot m$. Also, if she doesn’t reach a mouse before it ducks underground, she won’t be able to eat it. But she can eat a mouse if she reaches it precisely at the time it ducks underground.

Since all cats are efficient by nature, help Cartesian Cat determine what her minimum initial velocity must be if she hopes to eat all of the mice. Assume that she eats the mice in an optimal order.

-----Input-----
The first line of input contains a single positive integer, $n$ (where $1 \le n \le 15$), representing the number of mice. Each of the following $n$ lines describes a mouse given as three space-separated integers $x$, $y$, and $s$. This indicates that a mouse is located at $(x, y)$ and will duck underground at $t = s$ seconds. The values of $x$ and $y$ are in the range $[-1000, 1000]$ and $s$ is in the range $[1,10000]$. No two mice are at the same location. The last line of input contains a single floating-point number, $m$, specified to two decimal places, in the range $[0.75, 0.99]$, representing the multiplicative factor by which the cat slows down after eating a single mouse.

-----Output-----
Output the minimum initial velocity (in units per second) necessary for Cartesian Cat to allow her to eat all of the mice, given that she eats them in the optimal order. Your answer should be correct within a relative or absolute error of $10^{-3}$.

-----Examples-----
Sample Input 1:
1
3 4 2
.75
Sample Output 1:
2.4999999987500003

Sample Input 2:
2
0 100 10
0 -100 100
.80
Sample Output 2:
9.999999999000002
user: Aladin was walking down the path one day when he found the strangest thing: $N$ empty boxes right next to a weird alien machine. After a bit of fumbling around he got the machine to do something. The machine now accepts 4 integers $L$, $R$, $A$ and $B$. After that hitting the big red glowing button labeled “NE DIRAJ”1 causes the machine to go crazy and follow the next routine:
 - Set the number of stones in the box labeled $L$ to $A$ modulo $B$.
 - It procedes to fly to the box labeled $L+1$, and set the number of stones there to $(2\cdot A) \mod B$.
 - It procedes to fly to the box labeled $L+2$, and set the number of stones there to $(3\cdot A) \mod B$.
 - Generaly, it visits each box labeled between $L$ and $R$, and set the number of stones there to $( (X - L + 1)\cdot A) \mod B$, where $X$ is the box label.
 - After it visits the box labeled $R$. It settles down for further instructions.

During the game Aladin wonders what is the total number of stones in some range of boxes.

Write a program that simulates the device and answers Aladin’s questions.

-----Input-----
The first line contains two integers $N$ and $Q$ ($1 \leq N \leq 1000000000$) ($1 \leq Q \leq 50000$), number of boxes and number of queries.

The next $Q$ lines contain information about the simulation.

If the line starts with 1, than it follows the format “1 $L$ $R$ $A$ $B$” ($1 \leq L \leq R \leq N$) ($1 \leq A, B \leq 1000000$), meaning that Aladin keyed in numbers $L$, $R$, $A$ and $B$ in the device and allowed the device to do its job.

If the line starts with 2, then it follows the format “2 L R” ($1 \leq L \leq R \leq N$), meaning that Aladin wonders how many stones in total are ther stones are in boxes labeled $L$ to $R$ (inclusive).

-----Output-----
For each query beginning with 2 output the answer to that particular query. Queries should be processed in the order they are given in the input.First sample description

The boxes start containing $\{ 0, 0, 0, 0, 0, 0\} $, 0 stones in total. After that the device sets the stones to $\{ 1 \mod 2, 2 \mod 2, 3 \mod 2, 4 \mod 2, 5 \mod 2, 0\} $ = $\{ 1,0,1,0,1,0\} $, or 3 stones in total.

-----Examples-----
Sample Input 1:
6 3
2 1 6
1 1 5 1 2
2 1 6
Sample Output 1:
0
3

Sample Input 2:
4 5
1 1 4 3 4
2 1 1
2 2 2
2 3 3
2 4 4
Sample Output 2:
3
2
1
0