meetforever/GRPO_CL huggingface dataset:
user: George has won the lottery and, being a nice guy, has decided to spread the wealth around. However, monetary gifts can be taxed once they get over a certain size—the amount of tax depends on how much his friends have earned that year.

The amount of tax paid depends on tax bands. The bands start at zero. Each one takes a certain cut of income from the range of pre-tax income it covers. The final tax band applies to all income above its lower bound.

George is a savvy fellow and knows the number of tax bands, the amount of money each friend has earned and the amount he wants each friend to walk away with.

How much should George give to each friend before tax?

-----Input-----
 - One line containing an integer $B$ ($1 \le B \le 20$): the number of tax bands.
 - $B$ further lines, each containing two real numbers: $s_ i$ ($0 < s_ i \le 10^6$): the size in pounds of the $i^{th}$ tax band, and $p_ i$ ($0 \le p_ i \le 100$): the percentage taxation for that band.
 - One line containing a real number $P$ ($0 \le P \le 99.999$): the percentage tax on all income above other bands.
 - One line containing an integer $F$, ($0 < F \le 20$): the number of friends George wants to pay.
 - $F$ further lines, each containing two real numbers $e_ j$ and $m_ j$ ($0 \le e_ j \le 10^6$, $0 < m_ j \le 10^6$): the amount of money the $j^{th}$ friend has earned, and the amount of money they should receive after tax respectively.

Tax bands will be given in increasing order. All real numbers have at most $6$ decimals after the decimal point.

-----Output-----
 - $F$ lines, one for each friend specified in the input and in the same order.

Each line should contain one real number: the amount of money George will give to his friend, in order to arrive at the correct amount after they have paid tax.

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

-----Examples-----
Sample Input:
1
1000 0
20
3
0.0 500
999.5 500
1000.0 500
Sample Output:
500.000000
624.875000
625.000000
user: It is monsoon season, and your goldfish Orange is stuck at the bottom of a cave system in Thailand. Every hour, the water rises by $1$ meter. Thankfully, Orange has the ability to swim instantaneously from one location to another. However, he can’t hop over the sharp rocks in the way.

You trust that Orange is smart and will find the way out of the cave as soon as possible. To prepare Orange’s meal, you want to write a program to find out how long it will take for Orange to come back home.

You are given the cave system as an $N \times N$ grid where each location $(i, j)$ contains a nonnegative integer, $h_{i, j}$, representing the height of the rocks in meters at that location. Orange can only swim left, right, up, or down from one submerged location to another submerged location (but not diagonally).

A location $(i, j)$ is submerged if the water level is at least $1$ meter higher than $h_{i, j}$. Orange’s starting location will always be at $(0, 0)$, with $h_{0, 0} = 0$. The initial water level is $1$ meter so that Orange’s start location is submerged. The only exit in the cave is at location $(N - 1, N - 1)$. After how many hours can Orange find his way to the exit?

-----Input-----
The input consists of one integer, $N$, on the first line such that $2 \leq N \leq 100$, followed by $N$ lines with $N$ integers each, separated by spaces, such that each integer represents the height $h_{i, j}$ ($1 \le h_{i, j} \le 10^8$) of the rocks at location $(i, j)$.

-----Output-----
Output a single integer, which is the minimum number of hours that must pass before Orange can reach the exit.

-----Examples-----
Sample Input:
2
0 3
2 4
Sample Output:
4
user: A sequence of integers is said to Zigzag if adjacent elements alternate between strictly increasing and strictly decreasing. Note that the sequence may start by either increasing or decreasing. Given a sequence of integers, determine the length of the longest subsequence that Zigzags. For example, consider this sequence:

1 2 3 4 2

There are several Zigzagging subsequences of length $3$:

1 3 2        1 4 2        2 3 2        2 4 2        3 4 2

But there are none of length greater than $3$, so the answer is $3$.

-----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 input contains an integer $n$ ($1 \le n \le 1000000$) which is the number of integers in the list. Each of the following $n$ lines will have an integer $k$ ($1 \le k \le 1000000$).

-----Output-----
Output a single integer, which is the length of the longest Zigzagging subsequence of the input list.

-----Examples-----
Sample Input:
5
1
2
3
4
2
Sample Output:
3
user: The 3rd round of COCI is already here! In order to bet on predict the scores, we have assumed the following:
 - If contestant $A$ scored strictly more points than contestant $B$ in each of the first two rounds, then in the third round $A$ will score at least an equal amount of points as $B$.

Of course, in each round (including this one, the 3rd one) it is possible to score from 0 to 650 points. On the total ranking list, contestants are sorted descending according to the sum of points from all three rounds. The contestants with an equal sum share the same place and the next contestant gets the realistic following place. For example, contestants with sums equal to 1000, 1000, 900, 900 and 800 points win places 1, 1, 3, 3 and 5, respectively.

For each of the $N$ contestants, we know the number of points scored in the first and second round. Given the aforementioned assumption, determine the highest and lowest place each contestant can get on the total ranking list after three rounds of COCI.

-----Input-----
The first line of input contains an integer $N$ ($1 \leq N \leq 500000$), the number of contestants.

Each of the following $N$ lines contains two integers from the interval $[0, 650]$: the number of points each contestant won in the first and second round.

-----Output-----
For each contestant, in the order given in the input, output two integers per line: the required highest and lowest place they can get on the total ranking list.

-----Examples-----
Sample Input:
5
250 180
250 132
220 123
132 194
220 105
Sample Output:
1 3
1 3
3 5
1 5
3 5
user: You have a string $s$ and you may modify it by making long swaps of its letters. Two letters can be swapped if their positions differ by at least $k$. That is, you may swap the $i$-th letter with the $j$-th letter in $s$ if $|i - j| \geq k$. Is it possible to sort all the letters in $s$ increasingly, if you are allowed to swap any number of times (possibly zero)?

-----Input-----
The first line has a string $s$ ($2 \leq |s| \leq 100$) and an integer $k$ ($1 \leq k \leq |s| - 1$), separated by a single space. The string $s$ consists of only lowercase letters.

-----Output-----
If it is possible to sort the letters increasingly, output “Yes”. Otherwise output “No”.

-----Examples-----
Sample Input:
prognova 4
Sample Output:
Yes
user: Wallace the Weightlifting Walrus is training for a contest where it will have to lift $1000$ kg. Wallace has some weight plates lying around, possibly of different weights, and its goal is to add some of the plates to a bar so that it can train with a weight as close as possible to $1000$ kg.

In case there exist two such numbers which are equally close to $1000$ (e.g. $998$ and $1002$), Wallace will pick the greater one (in this case $1002$).

Help Wallace the Weightlifting Walrus and tell it which weight it will have to lift.

-----Input-----
The first line of the input contains the number of plates $n$ ($1 \leq n \leq 1000$). Each of the following $n$ lines contains one positive integer less than or equal to $1000$, denoting the weight of each plate.

-----Output-----
Output one integer, the combined weight closest to $1000$.

-----Examples-----
Sample Input:
4
900
500
498
4
Sample Output:
1002
user: Pierre is famous for his macarons. He makes round macarons, stored in square boxes of size $1\times 1$, and oval-shaped macarons, stored in rectangular boxes of size $1\times 2$ (or, rotated, in rectangular boxes of size $2\times 1$). For the purpose of a buffet, Pierre wishes to tile a rectangular table of size $N\times M$ with the two kinds of macarons, meaning that the table must be completely full, with no empty space left. The width $N$ of the table is small, for the guest to be able to grab the macarons easily, and the length $M$ of the table is large, to accommodate a huge number of guests. To keep the table pretty, the orientation of macarons should always be aligned with the sides of the table.

Pierre wishes to know how many ways there are to tile the table. Can you help him?Input

The input consists of the following integers:
 - the value of $N$, an integer, on the first line;
 - the value of $M$, an integer, on the second line.Limits

The input satisfies $1 \leq N \leq 8$ and $1 \leq M \leq 10^{18}$.Output

The output should consist of the total number of tilings, given modulo $10^9$, on a single line.

-----Examples-----
Sample Input:
2
2
Sample Output:
7
user: Peter returned from the recently held ACM ICPC World Finals only to find that his return flight was overbooked and he was bumped from the flight! Well, at least he wasn’t beat up by the airline and he’s received a voucher for one free flight between any two destinations he wishes.

He is already planning next year’s trip. He plans to travel by car where necessary, but he may be using his free flight ticket for one leg of the trip. He asked for your help in his planning.

He can provide you a network of cities connected by roads, the amount it costs to buy gas for traveling between pairs of cities, and a list of available flights between some of those cities. Help Peter by finding the minimum amount of money he needs to spend to get from his hometown to next year’s destination!

-----Input-----
The input consists of a single test case. The first line lists five space-separated integers $n$, $m$, $f$, $s$, and $t$, denoting the number of cities $n$ ($0 < n \le 50000$), the number of roads $m$ ($0 \le m \le 150000$), the number of flights $f$ ($0 \le f \le 1000$), the number $s$ ($0 \le s < n$) of the city in which Peter’s trip starts, and the number $t$ ($0 \le t < n$) of the city Peter is trying to travel to. (Cities are numbered from $0$ to $n-1$.)

The first line is followed by $m$ lines, each describing one road. A road description contains three space-separated integers $i$, $j$, and $c$ ($0 \le i, j < n, i \ne j$ and $0 < c \le 50000$), indicating there is a road connecting cities $i$ and $j$ that costs $c$ cents to travel. Roads can be used in either direction for the same cost. All road descriptions are unique.

Each of the following $f$ lines contains a description of an available flight, which consists of two space-separated integers $u$ and $v$ ($0 \le u, v < n$, $u \ne v$) denoting that a flight from city $u$ to city $v$ is available (though not from $v$ to $u$ unless listed elsewhere). All flight descriptions are unique.

-----Output-----
Output the minimum number of cents Peter needs to spend to get from his home town to the competition, using at most one flight. You may assume that there is a route on which Peter can reach his destination.

-----Examples-----
Sample Input:
8 11 1 0 5
0 1 10
0 2 10
1 2 10
2 6 40
6 7 10
5 6 10
3 5 15
3 6 40
3 4 20
1 4 20
1 3 20
4 7
Sample Output:
45
user: You may have heard of the “maximum subarray problem” from your university’s undergraduate algorithms course. The problem goes like this: You are given an array $A$ of $n$ integers, and the task is to find a contiguous subarray of $A$ whose sum is maximum. For example, in the array below:\[ \texttt{A := [-2, 3, 5, -7, 8, 13, -20, 14, 1]}, \]

the solution is the subarray from the second index (the number $3$) to the sixth index (the number $13$), with total sum $22$. The problem can be solved via divide and conquer, in $O(n \log n)$ time, or dynamic programming, in $O(n)$ time.

Being a strong student in algorithms yourself, you are of course familiar with both approaches, so we won’t bother explaining them here. However, your classmate Steve, a not-so-strong student in algorithms, has been bragging to you about the maximum subarray problem. He claims that not only can he solve the maximum subarray problem in linear time, he can even solve what he calls the “$k$-maximum subarray problem,” ALSO in linear time! What is the “$k$-Maximum subarray problem,” you ask? In a condescending tone, Steve explains: It is the natural generalization of the maximum subarray problem. Given an array $A$ of $n$ integers, you must find $k$ disjoint, contiguous subarrays of $A$ so that their total sum is maximum.

You’re not at all confident that Steve knows how to solve the “$k$-maximum subarray problem” in linear time, let alone that there even exists a linear time solution. You have gone to every algorithms lecture, however, a near-infinite improvement on Steve’s attendance, and so you believe if anyone is to be the authority on the problem, it probably should be you.

You decide to write a program to solve the “$k$-maximum subarray problem.” You can use this to verify Steve’s likely-incorrect algorithm. To simplify things, you make the program just output the value of an optimal solution. You can modify it later if you want the solution itself. Furthermore, you decide your algorithm is sufficiently efficient as long as its running time is bounded by a small polynomial in $n$ and $k$. You think coming up with an actually-correct linear time solution would probably take you awhile, and you’ve decided you only want to spend, say, a maximum of five hours on your code.

-----Input-----
The input will consist of two lines. On the first line are the integers $n$ and $k$, ($1 \leq k \leq n \leq 5000$). On the second line are $n$ integers, representing the array $A$. The integers in array $A$ will be between $-10^9$ and $10^9$, inclusive.

-----Output-----
Output the maximum possible total sum of $k$ disjoint contiguous subarrays of array $A$. Although the subarrays are required to be disjoint, a subarray may end at index $i$ and another subarray start at index $i+1$. No subarray is allowed to be empty.

-----Examples-----
Sample Input:
9 1
-2 3 5 -7 8 13 -20 14 1
Sample Output:
22
user: An elementary school class is going on a road trip by bus to a computer factory. However, the driver is really tired of having to drive fighting kids all day, so she suggested that the class should split up and go to the factory in groups. She has already observed which kids dislike each other and are likely to fight, so she wants to make sure they do not end up in the same group. Of course, driving everyone one-by-one is a waste of time and money, so the driver wants to minimize the number of groups she has to drive. In addition, the bus is pretty small, and may only fit up to $c$ kids at a time. 

You are to write a program which helps her with the task of making these groups. Given the number of kids and their enemies, find the minimum number of groups required, as well as a division of the class into this minimum number of groups

-----Input-----
The first line contains three integers $n$, $k$, and $c$ ($1 \le n \le 17$, $0 \le k \le \frac{n(n-1)}{2}$ and $1 \le c \le n$) – the number of kids, pairs of enemies, and the capacity of the bus, respectively. Then follow $n$ lines with the kids’ names. Each name consists solely of the characters A-Z and a-z, is non-empty, and at most $10$ characters long. Then follow $k$ lines, each containing a pair of space-separated names indicating a pair of kids that dislike each other. No pair of names appears twice, and no kid is their own enemy.

-----Output-----
On the first line, output the minimum number of groups, followed by one line per group containing the names of the children in that group (separated by spaces).

-----Examples-----
Sample Input:
2 0 1
Alice
Bob
Sample Output:
2
Alice
Bob
user: Chess has been a long time passion of Yraglac. He spents at least an hour per day playing chess matches online or reading chess tactics books. The game is played by two players on an $8 \times 8$ grid, with each player taking turns to move a piece.

In this problem, we will consider a simplified variant of chess consisting of two kinds of pieces: kings and rooks. Kings can move one square in any direction while rooks can move any number of squares up/down or left/right. Pieces can either move to an unoccupied square or to a square occupied by an opponent’s piece, which is captured and removed from play. A piece is threatened when an opponent’s piece is able to capture the piece in a single move. It is illegal to move one own’s king to a position where it is threatened (but it is legal to do so for a rook).

A player is considered to be in checkmate when it is their turn, their king is threatened, and all possible moves will result in the king still being threatened.

Yraglac is a skilled player and has managed to reach a situation where he has one king and one rook, while his opponent only has one king. Given that it is Yraglac’s turn, can he checkmate his opponent in a single move?

-----Input-----
The input consists of a single test case representing a chessboard. It will consist of eight lines with eight characters each, where “R” represents Yraglac’s rook, “K” represents Yraglac’s king, “k” represents the opponent’s king, and “.” represents an empty square. It is guaranteed that Yraglac will have exactly one king and one rook, while his opponent will only have one king. It is also guaranteed that no king will be threatened.

-----Output-----
Output “Yes” if Yraglac can checkmate his opponent in a single move, or “No” otherwise.

-----Explanation of Sample Input-----
In the first sample, Yraglac can checkmate his opponent by moving his rook to the bottom right square. In the second sample, there is no single move that will checkmate his opponent.

-----Examples-----
Sample Input:
........
........
........
........
.....K.k
........
........
R.......
Sample Output:
Yes
user: A town is often divided into zones, e.g, industrial zones, commercial zones, and residential zones. If some residential zone is very far from all commercial zones, then the people living there will have a long journey whenever they want to do some shopping, and this is undesirable.

The input will consist of an $n\times n$ grid of square zones. Each zone is labeled 1 (residential), 2 (industrial), or 3 (commercial). When travelling from one zone to another you are allowed to move north, east, south or west, and the distance travelled is the number of zone boundaries you traverse. So the distance between two adjacent zones is $1$, and the distance from the zone in square $(1,1)$ (the most south-westerly zone) to the zone in square $(2,3)$ is $3$ (one step east and two steps north). You may never move off the grid.

Your task is to find the longest distance one has to travel from a residential zone to get to the commercial zone closest to that residential zone.

-----Input-----
The first line of input contains an integer $n$, $2\leq n\leq 1500$, followed by $n$ lines of length $n$ giving the map of the city zones as an $n\times n$ matrix where each entry is 1, 2, or 3 depending on zone type. You can assume that the city has zones of all three types.

-----Output-----
Output a single integer $d$, the largest distance from a residential zone to its closest commercial zone.

-----Examples-----
Sample Input:
4
1223
2123
2213
3212
Sample Output:
3
user: You work at a clinic. The clinic factors in the waiting time when selecting patients to treat next. This approach was adopted to prevent patients from having to wait too long before being treated. Your task is to help the clinic perform $3$ types of queries:
 - Process a patient arrival to the clinic. The patient will have an arrival time $T$, a name $M$, and a severity $S$ that is accessed automatically by scanners at the entrance.
 - At time $T$, the doctor is ready to treat a patient. Every time this happens, the clinic will calculate priority values for every patient waiting in the clinic, and the patient with the highest priority value will be treated first. The priority value is computed as the following sum\[ S + K \cdot W \]

where $S$ is the severity value of the patient, $K$ is the constant that the clinic uses, and $W$ is the total time the patient has been waiting in the clinic. If there are multiple patients with that value, the patient with the lexicographically smallest name is treated next. Your program will announce the name of that patient.
 - At time $T$, the clinic receives a notification stating that, due to unfortunate circumstances, a patient with name $M$ has left the queue permanently. If no patient with name $M$ exists in the queue, it is always a false alarm and will be ignored by the clinic. Otherwise, the notification is guaranteed to be valid and should be processed accordingly.

-----Input-----
The first line of the input contains $2$ integers, $1 \leq N \leq 200000$, the number of queries to be processed, and $0 \leq K \leq 10000000$, the constant for the clinic. $N$ lines follow, each line beginning with an integer $Q$ where $Q$ = $1$, $2$ or $3$. $Q = 1$ denotes a query of the first type and will be followed by an integer $T$, a string $M$ and an integer $S$. $Q = 2$ denotes a query of the second type and will be followed by an integer $T$. $Q = 3$ denotes a query of the third type and will be followed by an integer $T$ and a string $M$. For all queries, $0 \leq T,S \leq 10000000$, and $T$ values are strictly increasing. $M$ is a non-empty alphanumeric string containing no spaces, and contains at most $10$ characters. All patients have unique names. There is at least one query of the second type.

-----Output-----
For each query of the second type, output the name of the patient who will be treated on a new line. If the clinic is empty, print the string “doctor takes a break” (without quotes) on a new line instead.

-----Subtasks-----
 - (28 Points): $1 \leq N \leq 10,000$. There is no query of type $3$. You can assume that $K = 0$.
 - (32 Points): $1 \leq N \leq 10,000$. There is no query of type $3$.
 - (20 Points): There is no query of type $3$.
 - (20 Points): No additional constraints.

-----Warning-----
The I/O files are large. Please use fast I/O methods.

-----Examples-----
Sample Input:
5 1
1 10 Alice 5
1 15 Bob 15
2 20
2 25
2 30
Sample Output:
Bob
Alice
doctor takes a break
user: Susan is making high-speed videos of falling coffee mugs. When analyzing the videos she wants to know how big the mugs are, but unfortunately they all got destroyed during filming. Susan knows some physics though, so she has figured out how far they moved between different video frames. The camera was filming at a speed of $70$ frames per second, which means that at frame $n$, counted from when the mug was released, the number of millimeters it has moved is $d=n^2$. The counting of the frames starts at $0$.

Susan thinks a certain mug is $D$ millimeters high, but to verify this she needs to find two frames between which the mug has moved exactly this distance. Can you help her do this?

-----Input-----
The input file contains the single positive integer $D\leq 200000$, the distance to be measured.

-----Output-----
Output two non-negative integers $n_1$ and $n_2$, the numbers of the frames that Susan should compare. They should fulfill $n_2^2-n_1^2=D$. If no two such integers exist, instead output “impossible”.

-----Examples-----
Sample Input:
88
Sample Output:
9 13