meetforever/GRPO_CL huggingface dataset:
user: The following puzzle was popularized by Martin Gardner’s book “My Best Mathematical and Logic Puzzles,” although it appears first in the monograph “On Problem-Solving” by the Gestalt psychologist Karl Dunker.

One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit.

The monk ascended the path at varying rates of speed, stopping many times along the way to rest and eat the dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting and meditation he began his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed.

Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day!

You can probably see why this is true - but can you write a program that computes the time at which the monk will be at the same spot during his ascent and descent?

-----Input-----
The input consists of a single test case. The first line contains two integers $a$ ($0 < a \le 5000$) and $d$ ($0 < d \le 5000$) denoting the number of segments during the ascent and descent, respectively. This is followed by $a$ lines, each containing two integers $h$ ($0 \le h \le 1000$) and $t$ ($0 < t \le 100$) denoting the positive change in elevation ($h$) the monk gained during this segment and the time it took ($t$). If the monk rests and eats during a segment, $h$ will be $0$.

This is followed by $d$ lines, each containing two integers $h$ ($0 \le h \le 1000$) and $t$ ($0 < t \le 100$) denoting the change in elevation ($h$) the monk descended during this segment and the time it took ($t$). If the monk rests and eats during a segment, $h$ will be $0$.

The total change in elevation during the ascent and descent is guaranteed to be the same, i.e., the sum of the changes in elevation for the ascending segments is the same as the sum for the descending segments.

-----Output-----
Output a single floating point number, the earliest point in time at which the monk occupies the same spot during his climb and his descent. The monk starts his ascent and his descent at time $0$ on both days.

Your answer will be considered correct if its absolute or relative error does not exceed $10^{-5}$.

-----Examples-----
Sample Input 1:
1 1
10 11
10 10
Sample Output 1:
5.238095

Sample Input 2:
3 1
4 2
0 3
6 3
10 7
Sample Output 2:
4.200000
user: After last year’s edition of the BAPC, you are still stuck in Delft. In order to participate again this year, you are going to Amsterdam by bus. During the journey you look out of the window and look for traffic signs that point in the direction of Amsterdam. To your surprise, you notice that the bus is never taking the roads that are pointed out by the signs!

You think that the bus company might have chosen a route such that, at no intersection, the bus goes in the direction that is pointed to by the signs. Your friends, however, find this very unbelievable, and don’t think this is possible. Can you figure out whether there exists a bus route that satisfies this requirement? Note that a bus route never visits the same place twice.

A traffic sign pointing in the direction of the shortest route to Amsterdam is placed at every intersection. You may assume that the input graph is both simple and connected, and that there is a unique optimal route to Amsterdam from every intersection.

-----Input-----
 - A single line containing two integers: $n$ ($2 \le n \le 10^5$), the number of intersections, and $m$ ($1 \le m \le 10^6$), the number of undirected roads that connect the intersections. The intersections are numbered from $0$ to $n-1$. Delft is denoted by intersection $i=0$ and Amsterdam is denoted by intersection $i=1$.
 - $m$ lines that specify the roads
 - A road is specified by three integers, $a_ i$, $b_ i$ ($0 \leq a_ i, b_ i < n$ and $a_ i \ne b_ i$) and $d_ i$ ($0 \le d_ i \leq 500000$), where $a_ i$ and $b_ i$ are ids of the two intersections that are connected by this road and $d_ i$ is the distance that the bus has to travel to get from $a_ i$ to $b_ i$ or vice versa.

-----Output-----
As output, give one of the following:
 - A path from Delft to Amsterdam that satisfies the requirements, in case such a path exists.
 - A path is specified by a single line containing an integer $k$, the length of the path, followed by $k$ integers $p_ i$ that specify the intersections along the path in the order in which they are crossed, with $p_0 = 0$ and $p_{k-1}=1$.
 - The text “impossible”, if such a path does not exist.

-----Examples-----
Sample Input 1:
4 5
0 2 5
2 1 5
0 3 10
3 1 20
3 2 5
Sample Output 1:
3 0 3 1

Sample Input 2:
4 3
0 1 10
1 2 20
2 3 30
Sample Output 2:
impossible
user: As a New Yorker you are always very busy. Apart from your long work day you tend to have a very long list of errands that need to be done on any particular day. You really hate getting up early so you always end up going over your to-do list after work, but this is starting to seriously hurt your free time.

One day you realize that some of the places you have to go by lie on your walk to the office, so you can visit them before work. The next day you notice that if you take a slightly different route to work you can run most of your errands without increasing the length of your route. Since errands take a negligible amount of time, you don’t even have to get up any earlier to do this! This nice little side effect of the grid-like New York streets gets you thinking. Given all the locations of your errands on the New York grid, how many can you visit on your way to work without getting up any earlier?

The New York grid is modelled with streets that are parallel to the $x$-axis and avenues that are parallel to the $y$-axis. Specifically, there is a street given by $y=a$ for every $a\in \mathbb Z$, and there is an avenue given by $x=b$ for every $b\in \mathbb Z$. It is given that an errand always takes place on an intersection between a street and an avenue. Since you walk to your work, you can use all roads in both directions.

-----Input-----
 - The first line contains an integer $0\leq n\leq 10^5$, the number of errands you have to run that day.
 - The next line contains four integers $0\leq x_ h, y_ h, x_ w, y_ w\leq 10^9$ corresponding to the locations of your house and workplace.
 - The next $n$ lines each contain two integers $0\leq x_ i,y_ i\leq 10^9$, the coordinates of your $i$th errand.

-----Output-----
Output a single line, containing the number of errands you can run before work without taking a longer route than necessary.

-----Examples-----
Sample Input 1:
3
0 0 6 6
5 4
2 6
3 1
Sample Output 1:
2

Sample Input 2:
5
2 1 0 0
0 0
0 1
2 0
2 1
3 1
Sample Output 2:
3
user: Everyone’s favorite character and puppet-maker Geppetto has opened a new pizza place, the best in town. Geppetto is trying to make the best pizza possible, but at the same time he doesn’t want to have a small selection of pizzas.

He makes his pizzas out of $N$ ingredients marked with numbers from $1$ to $N$. All that would be simple if he could mix any ingredient with every ingredient on the pizza, but unfortunately, that is not the case. Sometimes some ingredients cannot mix and that creates additional complications for our pizza master.

There are $M$ pairs of ingredients that cannot be on the same pizza at the same time. Given these restrictions, Geppetto wants to know how many different pizzas he can make. Help him answer this question. Two pizzas are considered different if there is an ingredient of index $i$ that is on one pizza, but not on the other.

-----Input-----
The first line of input contains two integers $N$ and $M$ ($1 \leq N \leq 20$, $0 \leq M \leq 400$). Each of the following $M$ lines contains two different numbers $a$ and $b$, they represent the prohibition of mixing ingredients marked with $a$ and $b$ on the pizza. ($1 \leq a, b\leq N$). All pairs of ingredients are not necessarily distinct, some pair could occur multiple times.

-----Output-----
The first and only line of output must contain the number of different pizzas given the restrictions in the task.

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

Sample Input 2:
3 0
Sample Output 2:
8
user: Suppose you are given two integers, $m$ and $n$. You are also given a list of $n$ distinct integers $x_1, x_2, \ldots , x_ n$, with $0\! \le \! x_ i\! \le \! 2^m{-}1$. For each number $y$ from $0$ to $2^m{-}1$, you’ve found the number $p_ y$ such that $x_{p_ y}$ has a maximum bitwise-$\operatorname {XOR}$ with $y$. That is, $y\! \oplus \! x_{p_ y}\! >\! y\! \oplus \! x_ i$ for all $i\! =\! 1..n, i\! \neq \! p_ y$ ($\oplus $ means bitwise-$\operatorname {XOR}$).

Now, consider the reverse problem. Given $m$, $n$, and the sequence $p_0, p_1, \ldots , p_{2^m{-}1}$, count the number of sequences of distinct integers $x_1, x_2, \ldots , x_ n$ that could have generated that $p$ sequence from the above algorithm. Two $x$ sequences are different if there is some $i$ such that $x_ i$ in one sequence is different from $x_ i$ in the other sequence. Output this count modulo $10^9{+}7$.

-----Input-----
Each test case will begin with a line with two space-separated integers $m$ ($0\! \le \! m\! \le \! 16$) and $n$ ($1\! \le \! n\! \le \! 2^m$), where $2^m$ is the length of the $p$ sequence, and $n$ is the length of the $x$ sequences.

Each of the next $2^m$ lines will contain a single integer $p$ ($1\! \le \! p\! \le \! n$). These are the values of the sequence $p_0, p_1, \ldots , p_{2^m{-}1}$, in order. Every value from $1$ to $n$ inclusive will appear at least once.

-----Output-----
Output a single integer, which is the number of sequences $x_1, x_2, \ldots , x_ n$ which could have generated the sequence $p_0, p_1, \ldots , p_{2^m{-}1}$ from the above algorithm, modulo $10^9{+}7$.

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

Sample Input 2:
2 3
1
2
1
3
Sample Output 2:
0
user: Johanna knows mind reading magic, or so she says. Her new trick consists of lining up $N$ decks of cards, each deck having $K$ cards numbered from $1$ to $K$. She asks you to think of a number $T$ between $1$ and $N \cdot K$ and to focus your thoughts on it. Then, by scanning the energy profiles emitted from your brain activity, she carefully picks one card from each of the $N$ decks. Magically, the sum of the numbers on the $N$ picked cards is exactly the number that you were thinking of! After staring at her in disbelief for quite a while, you suspect it might be a trick she pulls on many people, and that she just picks the cards at random and happened to get it right with you just by chance.

You start wondering just how large that chance was. You could easily compute the number of ways to pick one card from each deck, but how many of these ways have the correct sum?

-----Input-----
The first line contains three space-separated integers $1 \le N \le 100$, $1 \le K \le 50$, $1 \le T \le N \cdot K$ – the number of decks, the number of cards in each deck, and the number you were thinking of, respectively.

-----Output-----
Output a single integer – the number of ways Johanna could have picked a card from each deck, such that the picked cards sum up to $T$. Since this number can be very large, output it modulo $1000000009$.

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

Sample Input 2:
5 5 25
Sample Output 2:
1
user: The First Universal Bank of Denview has just been robbed! You want to catch the robbers before they leave the state.

The state of Calirado can be represented by a rectangular $n$-by-$m$ grid of characters, with the character in each grid cell denoting a terrain type. The robbers begin within the cell marked ‘B’, indicating the Bank of Denview. They will then travel across the state by moving from grid cell to grid cell in the four cardinal directions (left, right, up, down). (Note that the robbers pass only through grid edges, and not corners.) If the robbers manage to leave the state (by crossing any boundary edge of the grid) they will go into hiding, never to be seen again. You must stop this.

To catch the robbers, you can set up barricades. Barricades are placed inside a grid cell, and prevent the robbers from traveling into the cell (from any direction). Each grid square consists of a different type of terrain, with different cost for placing a barricade. You cannot place a barricade on the bank (‘B’) or on any cell containing a dot (‘.’), though the robbers can travel freely through these cells. Every other cell will contain a lowercase English letter, indicating a terrain type.

Find the cheapest way to prevent the robbers from escaping Calirado.

-----Input-----
The first line contains three integers $n$, $m$, and $c$ ($1 \le n, m \le 30$, $1 \le c \le 26$): the dimensions of the grid representing Calirado, and the number of different terrain types. Then follows $m$ lines of exactly $n$ characters each: the map of Calirado. Each character is either ‘B’, ‘.’, or one of the first $c$ lowercase letters of the English alphabet. Calirado is guaranteed to contain exactly one bank. After the grid, there is a line containing $c$ space-separated integers $1 \leq c_ i \leq 100000$, the costs of placing a barricade on a grid cell of each terrain type. $c_1$ is the cost for terrain type ‘a’, $c_2$ is the cost for ‘b’, and so forth.

-----Output-----
Print one integer, the minimum total cost of the barricades that you need to place to prevent the robbers from escaping. If there is no way to prevent the robbers from escaping, print -1 instead.

In the first example, the minimum cost is to barricade the central three squares on each side of the bank for a total cost of $12$.

In the second example, since the bank is on the border, there is no way to prevent the robbers from escaping the state.

In the third example, we must prevent the robbers from leaving the bank to the top, bottom, and right, or else we cannot prevent them from leaving the state. To the left, however, it is cheaper to allow passage through the ‘b’ cell, and then barricade in each of the three directions from there. The total cost is $7 + 5 + 7 + 3(1) = 22$.

-----Examples-----
Sample Input 1:
5 5 1
aaaaa
a...a
a.B.a
a...a
aaaaa
1
Sample Output 1:
12

Sample Input 2:
2 2 1
aB
aa
1
Sample Output 2:
-1
user: Prof. Peter decided to surprise his class by organizing a new lottery with a very peculiar system.

He starts by announcing a small positive number $M$. Afterwards, each student is going to secretly write an integer from the set $\{ 1, ..., M\} $ on a slip of paper that is then folded. After everyone has selected a number, they reveal all choices and whoever wrote down the lowest unique number is the winner! For instance, if there are only three students in the class, two of them select number $2$ and the other selects number $5$, then the student who chose number $5$ is the winner.

The lottery was a success, but Prof. Peter is now wondering what strategy his students should have used. If everyone follows the same optimal randomized strategy, with which probability should each number be chosen so that they maximize their chances of winning? A strategy is optimal if, when everyone is following it, then no individual student can improve his winning probability by selecting a different strategy. Can you help Prof. Peter?

-----Task-----
Given $N$, the number of students in the class, and $M$, the largest number they can write down, determine the optimal randomized strategy (followed by all students). That is, determine the probability of selecting each number between $1$ and $M$.

-----Input-----
There is one line with two integers: $N$, which is the number of students in the class, and $M$, which is the largest integer each student can choose.

-----Constraints-----

$3$

$\leq $

$N$

$\leq $

$7$

Number of students in the class

$1$

$\leq $

$M$

$\leq $

$N+1$

Maximum number that can be selected

-----Output-----
The output consists of $M$ lines. Line $k$ should have the probability of selecting number $k$.

The result will be considered correct as long as the absolute error does not exceed $10^{-3}$.

-----Examples-----
Sample Input 1:
3 3
Sample Output 1:
0.46410
0.26795
0.26795

Sample Input 2:
7 1
Sample Output 2:
1.00000
user: Liam is planning out his route for an upcoming nature hike. Unfortunately, the region that he plans on hiking through is notoriously muddy, so in order to prevent his floral shirt from getting covered with mud, he has decided to pick a route for which the maximum depth of mud encountered is minimized.

The terrain is modelled as a two-dimensional grid, with nonnegative integers giving the depth of the mud in each cell in micrometers (millionths of a meter). The hike must begin at one of the cells in the leftmost column of the grid and finish at one of the cells in the rightmost column of the grid. Liam is allowed to move either north, south, east, or west, but he cannot travel diagonally.

-----Input-----
The first line of input contains two space-separated integers, $r$ and $c$ $(2 \leq r, c \leq 1000)$, indicating the number of rows and the number of columns in the grid, respectively. Each of the next $r$ lines contains $c$ space-separated integers, indicating the depth of mud in the cells of the corresponding grid row from left to right. Each depth is measured in micrometers and is given as an integer in the interval $[0,1000000]$.

-----Output-----
The output should contain a single integer, the depth of the deepest mud that Liam is forced to hike through if his route minimizes this value.

-----Examples-----
Sample Input 1:
5 4
2 1 0 8
3 7 3 5
3 1 2 4
9 0 4 6
5 3 2 3
Sample Output 1:
3

Sample Input 2:
2 2
3 0
1 2
Sample Output 2:
2
user: An airline called Divided Airlines has recently made the news due to their tendency to overbook their flights rather aggressively. For some flights, they even resorted to dragging passengers out from the plane! This was of course not very popular, so they decided to “resolve” the issue by making the seating assignments very chaotic (airlines do like unnecessary complexity). 

A particular flights has $n$ passengers. The seats are divided into $r$ rows each containing $c$ seats. Every passenger $i$ is assigned to some particular seat located at row $a_ i$ and column $b_ i$. However, some passengers may be assigned to the same seat.

Of course, passengers are usually okay with sitting somewhere else than their assigned seat, but they may still want to be somewhat close to their original seat. Perhaps they want to be able to speak to their friends, or sit close to their overhead luggage. More specifically, passenger $i$ accepts sitting at most $s_ i$ rows away from the row on their ticket.

Due to budget reasons, you decided to travel on a Divided flight. As expected, all the passengers assigned to an overbooked seat started to fight with each other, moving around in complex ways and causing a long delay. You proposed a fair resolution: you will construct a seat assignment which takes into account how far the passengers accepts to sit from their assigned seats so that as many passengers as possible get a seat. Now, all that remains is to actually find this assignment.

-----Input-----
The input consists of:
 - one line with the integers $n$, $r$ and $c$ ($1 \le n, r, c \le 10^5$), the number of passengers, rows and columns in the flight.
 - $n$ lines with the integers $a_ i, b_ i$ and $s_ i$ ($1 \le a_ i \le r$, $1 \le b_ i \le c$, $0 \le s_ i \le r$). The $i$’th line has the assigned row $a_ i$ and column $b_ i$, and maximum distance $s_ i$ of the $i$’th passenger. The maximum distance is given in rows.

-----Output-----
Output the maximum number of passengers that can be assigned a seat in an optimal assignment.

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

Sample Input 2:
3 3 1
1 1 0
1 1 1
1 1 2
Sample Output 2:
3
user: Summer in Texas can be very hot. But Texans are tough, and in a weird way some of them enjoy the heat; for even the heat is bigger in Texas. But the heat can be quite uncomfortable for computer science students who are new to Texas and are not used to the sweating it can cause.

These students want to minimize the amount they sweat while walking from their dormitory to class. When they enter the sunshine, they begin sweating at the rate $r_0$ gallons/hour (where $r_0 > 0$). The longer they stay in the sun, the more they sweat. They sweat at a rate proportional to the square of how long they have been continuously exposed to the sun. Put another way, if they are exposed to the sun for twice as long, they sweat four times as much. But if they find a shady spot along the way, their continuous sun exposure is broken and they stop sweating immediately. They can then sit in the shade and cool off completely before continuing on their journey to class. Of course, leaving the shade means they begin sweating again, but starting at the lower rate $r_0$ (since they have cooled off).

Write a program that helps a student find a path from the dormitory to class which minimizes the total amount of sweat she expends.

-----Input-----
Input describes the locations of shady spots on campus as well as the student’s dormitory and class locations. Input begins with a line containing an integer $0 \leq n \leq 2500$, the number of shady spots. Each of the next $n$ lines contains a pair of integers $x~ y$ specifying the coordinates of a shady spot. No two shady spots have the same coordinates. Following the shady spots are two more lines in the same format which specify the coordinates of the student’s dormitory and class, respectively.

-----Output-----
Print a path the student can take to get from her dormitory to class. The path should minimize the total sweat produced along the whole path. Print the path as indexes of the shady spots (from the order given in the input, with the first shady spot having index 0). If the best path contains no shady spots, output a single ‘-’. If there are multiple paths that minimize the total sweat, print any one of them.

-----Examples-----
Sample Input 1:
3
1 1
2 -2
5 -1
0 0
9 0
Sample Output 1:
1
2

Sample Input 2:
6
8 2
4 0
8 0
4 -1
7 -1
6 -2
2 1
9 2
Sample Output 2:
1
3
5
4
2