meetforever/GRPO_CL huggingface dataset:
user: Peter is bored during his operating systems class, so he draws doodles on a sheet of paper. He feels like drawing abstract art using his ruler: he draws line segments by choosing two points in the plane and connecting them. Lots of them.

Can you write a program that counts the number of distinct points at which the line segments he drew intersect or touch?

-----Input-----
The first line in the input contains an integer $n$ ($1 \le n \le 1000$) which is the number of lines. The following $n$ lines contain four integers $x_0 \ \ y_0 \ \ x_1 \ \ y_1$ ($-1000000 \le x_0, \ y_0, \ x_1, \ y_1 \le 1000000$). Lines have non-zero length, i.e., the two points will be distinct: $x_0 \ne x_1$ or $y_0 \ne y_1$ or both.

-----Output-----
Output the number of distinct points for which there is at least one pair of line segments that intersects or touches at this point. If there are infinitely many such points, output -1.

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

Sample Input 2:
3
5 2 7 10
7 4 4 10
2 4 10 8
Sample Output 2:
1

Sample Input 3:
3
2 1 6 5
2 5 5 4
5 1 7 7
Sample Output 3:
1

Sample Input 4:
2
-1 -2 -1 -1
-1 2 -1 -1
Sample Output 4:
1
user: A Hill Number is a positive integer, the digits of which possibly rise and then possibly fall, but never fall and then rise. For example:12321 is a hill number.

12223 is a hill number.

33322111 is a hill number.

1232321 is not a hill number.


Given a positive integer, if it is a hill number, print the number of positive hill numbers less than or equal to it. If it is not a hill number, print -1.

-----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 integer $n$ ($1 \le n \le 10^{18}$).

-----Output-----
Output a single line with a single integer. If the input is a hill number, then output the number of hill numbers less than or equal to it. If the input is not a hill number, then output -1.

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

Sample Input 2:
55
Sample Output 2:
55

Sample Input 3:
101
Sample Output 3:
-1

Sample Input 4:
1234321
Sample Output 4:
94708
user: Archimedes is the name of a new 2D video game, which is very simple: the player hits start, at which point their avatar moves along an Archimedean spiral starting from the origin. When they hit a button, the avatar detaches from the spiral in the direction it is currently moving. The goal is to hit a target. The avatar’s path after it detaches must not intersect with any part of the spiral.

To help study this game, you’re asked to write a program that given the spiral’s angular velocity computes the point at which the player would need to press the button to release the avatar from the spiral in order to hit the target.

An Archimedean spiral is created when a point moves with constant velocity along a line that rotates with constant angular velocity around the origin. When expressed in polar coordinates $(r, \phi )$, an Archimedean spiral has a simple formulation: $r = b \ \phi $ where $r$ is the distance from the origin and $\phi $ is the angle between the point, origin, and the unit vector $(1, 0)$. $b$ is a constant that determines the distance between successive rings of the spiral.

-----Input-----
The input consists of a single line with $3$ real numbers $b$, $t_ x$, and $t_ y$, denoting the parameter $b$ ($0.01 \le b \le 10$) of the spiral described by $r = b \ \phi $ and the $x, y$ coordinates of the target $T = (t_ x, t_ y)$, restricted by $-10000 \le t_ x, t_ y \le 10000$. It is guaranteed that $\sqrt{t_ x^2+t_ y^2} > 2 \pi b$, i.e., the avatar will stay on the spiral for at least one full $360$ degree turn. It is also guaranteed that the distance from point $T$ to the closest point on the spiral will be greater than $10^{-3}$. There may be up to $12$ significant digits in $b$, and up to $3$ digits after the decimal point in $t_ x$ and $t_ y$.

-----Output-----
Output the $x, y$ coordinates of the point on the spiral where the avatar should leave the spiral, continue in the direction it is moving, and hit the target without intersecting the spiral.

Your answer will be considered correct if the absolute or relative error of both $x$ and $y$ does not exceed $10^{-5}$.

-----Examples-----
Sample Input 1:
0.5 -5.301 3.098
Sample Output 1:
-1.26167861 3.88425357

Sample Input 2:
0.5 8 8
Sample Output 2:
9.21068947 2.56226688

Sample Input 3:
1 8 8
Sample Output 3:
6.22375968 -0.31921472

Sample Input 4:
0.5 -8 8
Sample Output 4:
-4.36385220 9.46891588
user: You own a disco called the Boogie Always Persists Club. The club is famous for its multiple interconnected rooms to twist and shout in. The rooms and the corridors between them form a maze-like structure and for added bonus you have made all the corridors one-way. However, it turns out not everyone is as happy with your club as you are. Recently the fire safety inspectors came by and they were not amused by what they saw: if a fire were to break out in one of the rooms, people would have great difficulty finding the fire exits and might even start running around in circles! They find this completely unacceptable and order you to improve things as quickly as possible. They insist that you have to make sure that no one can run around in circles in the club by removing some of the corridors between the rooms.

You, on the other hand, want to retain the attractiveness of the rooms. You do not want to remove too many corridors, because then people will no longer visit your club. You decide that at most half of the corridors may be removed.

Given the layout of the club, remove at most half of the corridors so that no cycles remain.

-----Input-----
 - One line containing the number of rooms $1\leq n\leq 10^5$ and the number of corridors $0\leq m\leq 2\cdot 10^5$.
 - Then follow $m$ lines, each containing two different $1$-based integers $u$ and $v$ indicating a corridor from room $u$ to room $v$. There will be no corridor from a room to itself, nor will there be more than one corridor from one room to any other single room.

-----Output-----
 - On the first line, print a single integer $0\leq r\leq m/2$, the number of corridors to be removed.
 - Then print $r$ lines containing the $1$-based indices of the corridors that need to be removed to ensure that dancers cannot go around in circles in the disco anymore.

If there are multiple valid solutions, you may output any one of them.

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

Sample Input 2:
3 3
1 2
2 3
3 1
Sample Output 2:
1
1

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

Sample Input 4:
4 5
1 2
2 3
2 4
3 1
4 1
Sample Output 4:
2
4
5
user: Harry the Hamster lives in a giant hamster cage. Inside the cage there is a set of $n$ plastic balls connected by unidirectional hamster tubes of varying lengths. Harry is currently in ball $s$ and his bed is in ball $t$.

Being a simple hamster, the hemispheres of Harry’s brain are not so great at communicating with each other and have a mind of their own. Harry’s left hemisphere, usually being active when Harry is in the hamster wheel, likes running for as long as possible. Harry’s right hemisphere, rarely active at all, would like to go to sleep as soon as possible. Together, Harry’s hemispheres will be navigating Harry through the maze of tubes, in each ball deciding which of the outgoing tubes to follow.

Harry’s hemispheres get really tired after making a decision and then need to rest a bit, so they cannot make two decisions in a row. Thus, they make decisions on which tube to take in alternating turns, with the left hemisphere going first. So starting in ball $s$, Harry’s left hemisphere will decide on a tube to follow, ending in some ball $u$, where Harry’s left hemisphere will rest and Harry’s right hemisphere will pick an outgoing tube, et cetera.

Counterintuitively, the hemispheres are familiar with the entire hamster cage and can plan arbitrarily far ahead. Assuming both hemispheres make optimal decisions, how long will it take for Harry to reach his bed? It is guaranteed that each ball has at least one outgoing tube, except the ball containing Harry’s bed which has none (there Harry will rest easily). There are no tubes connecting a ball to itself, but there may be multiple tubes going from one ball to another.

-----Input-----
 - On the first line are four space-separated integers: the number of plastic balls $1 \leq n \leq 10^5$, the number of tubes $0 \leq m \leq 2 \cdot 10^5$, and the locations of Harry and his bed $0 \leq s, t < n$.
 - Then $m$ lines follow, each containing three space-separated integers describing a single tube: the ball in which the tube starts $0 \leq a_ i < n$, in which it ends $0 \leq b_ i < n$ and the time it takes to traverse $1 \leq w_ i \leq 10^4$. Note that each tube can only be traversed in one direction.

-----Output-----
Print the time it takes for Harry to reach his bed, or the string infinity if Harry is doomed to roam the tubes forever.

-----Examples-----
Sample Input 1:
4 5 0 3
0 1 1
1 2 2
2 0 4
2 3 1
2 3 3
Sample Output 1:
11

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

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

Sample Input 4:
3 3 1 2
0 1 1
1 0 1
1 2 1
Sample Output 4:
infinity
user: Baby Timmy has a pool of balls that he plays with. The pool is made up of balls with $n$ different colors with a certain number of each color. Baby Timmy finds it interesting to order these balls in a line given certain conditions he made. Timmy has two conditions when playing his own game:
 - Balls of certain colors may not be next to each other
 - One particular sequence of colors which Timmy likes most must appear as many times as possible

Can you compute the total number of different ways in which Timmy can arrange his balls?

For instance, suppose Timmy has $2$ red, $1$ yellow, $2$ green, and $1$ blue ball(s). He doesn’t like for red or yellow balls to be next to each other, and his favorite sequence is “green blue.” The following six arrangements meet the requirements:

red green blue red green yellow 
red green blue yellow green red 
red green red green blue yellow 
red green yellow green blue red 
yellow green blue red green red 
yellow green red green blue red 

This arrangement corresponds to Sample Input 1.

Note that Timmy insists on including his favorite sequence as often as the available balls allow, even if that means not being able to complete the arrangement of balls at all.

-----Input-----
The input consists of a single test case with three lines. The first line contains an integer $n$ ($2 \le n \le 50$), which describe the number of different colors. The remaining $n$ integers on that line denote how many balls Timmy has of each color (colors are numbered $1$ through $n$ and their frequencies appear in order). The number of balls he has of each color is between $1$ and $50$, inclusive.

The second line of input describes which colors Timmy does not want next to each other. The first integer $k$ ($0 \le k \le n$) gives the number of colors. This is followed by $k$ integers $c_ i$ ($1 \le c_ i \le n$) denoting the colors that prevent balls having any of these colors from being next to each other. Each $c_ i$ is unique.

The third line of input describes the sequence Timmy likes most. This first integer $l$ ($0 \le l \le n$) describes the length of this sequence, and the following $l$ integers $s_ i$ ($1 \le s_ i \le n$) describe the sequence that must appear as often as possible in the arrangement. Each $s_ i$ is unique and the sets $\{ c_ i \} $ and $\{ s_ i \} $ do not intersect.

-----Output-----
Output the number of arrangements Timmy can make that satisfy the given conditions. Since the number can be large, output its value modulo $1000000007$.

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

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

Sample Input 3:
3 2 2 3
1 1
2 2 3
Sample Output 3:
18

Sample Input 4:
3 1 2 3
2 1 2
0
Sample Output 4:
12
user: Theta has been learning about powers of $2$ in school. She notices that some numbers when written out contain powers of $2$ in their digit representation: for instance, $12560$ contains $256$ which is a power of $2$. She has been wondering how many such numbers there are.

Can you write a program that counts how many numbers contain a given power of $2$?

-----Input-----
The input consists of a single line with two integers $n$ and $e$ ($0 \le n \le 9 \cdot 10^{18}, 0 \le e \le 62$).

-----Output-----
Output a single integer that is equal to the number of distinct integers $k$ ($0 \le k \le n$) whose decimal representation contains the digits of $2^e$ as a substring.

-----Examples-----
Sample Input 1:
1000000 1
Sample Output 1:
468559

Sample Input 2:
1000000 5
Sample Output 2:
49401

Sample Input 3:
1000000 16
Sample Output 3:
20

Sample Input 4:
9000000000000000000 62
Sample Output 4:
1
user: Two players are playing a game with a set of $K$ six-sided dice. One player calls out a number in the range $K\ldots 6K$ and the other tries to roll that number. After the first roll, the player is allowed to pick up any number ($0\ldots K$) of dice and re-roll them.

Given the number of dice, the target number the player wants to roll, and the set of numbers the player obtained on the first roll, what number of dice should the player pick up to maximize their chances of getting the target number on the second roll?

-----Input-----
Input begins with a line containing $2$ integers, $K$, the number of dice, and $T$, the target number. The bounds are $2 \leq K \leq 24$ and $K \leq T \leq 6K$.

The next line contains $K$ integers, indicating the numbers that were rolled on each of the dice on the first roll. All will be integers in the range $1\ldots 6$.

-----Output-----
Print a single line containing an integer denoting the number of dice that the roller should pick up and re-roll in order to maximize the chances of getting an overall sum of $T$. (The roller will be able to choose which dice to pick up, but you are only asked to print the number of dice, not which ones.)

If there are more than one numbers of dice that can be picked up to achieve the same probability of getting to $T$, print the smallest such value.

-----Examples-----
Sample Input 1:
3 9
5 4 1
Sample Output 1:
1

Sample Input 2:
4 13
2 2 2 2
Sample Output 2:
3

Sample Input 3:
18 90
1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
Sample Output 3:
12
user: Today everyone is talking about big data. As a renowned data provider in the area, you now have $N$ precious pieces of data. You want to sell all $N$ pieces of data to $N$ customers. You can sell multiple pieces of data (including all and none) to a single customer. However, each piece of data must be sold to exactly one customer (you cannot split a single piece of data and sell it to multiple customers).

In your previous transactions you simply sold bigger data for higher prices. Unfortunately, that did not work quite well and your customers were a bit unsatisfied about the value of the data they got. Obviously, the value of data is not necessarily positively correlated with its volume.

This time you decide to do something different. After many meetings you finally sign a new contract with your customers. You number the $N$ pieces of data from $1$ to $N$ and figure out that the $i^\textrm {th}$ piece of data has a special property $S_ i$. The new contract specifies that each customer shall pay you an amount of money that is equal to the number of distinct prime factors in the sum of $S_ i$’s of the data pieces sold to him/her.

Based on the new contract, what is the maximum revenue you can achieve by selling all $N$ pieces of data?

-----Input-----
The first line has a single integer $N$ ($1 \leq N \leq 14$). The second line has $N$ positive integers, $S_1, S_2, \ldots , S_ N$. These integers are no larger than $1000$.

-----Output-----
Output the maximum revenue you can achieve by selling all $N$ pieces of data.

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

Sample Input 2:
3
4 7 8
Sample Output 2:
3

Sample Input 3:
5
2 3 4 5 8
Sample Output 3:
5
user: “Drat!” cursed Charles. “This stupid carry bar is not working in my Engine! I just tried to calculate the square of a number, but it’s wrong; all of the carries are lost.”

“Hmm,” mused Ada, “arithmetic without carries! I wonder if I can figure out what your original input was, based on the result I see on the Engine.”

Carryless addition, denoted by $\oplus $, is the same as normal addition, except any carries are ignored (in base $10$). Thus, $37 \oplus 48$ is $75$, not $85$.

Carryless multiplication, denoted by $\otimes $, is performed using the schoolboy algorithm for multiplication, column by column, but the intermediate additions are calculated using carryless addition. More formally, Let $a_ m a_{m-1} \ldots a_1 a_0$ be the digits of $a$, where $a_0$ is its least significant digit. Similarly define $b_ n b_{n-1} \ldots b_1 b_0$ be the digits of $b$. The digits of $c = a \otimes b$ are given by the following equation:\[ c_ k = \left( a_0 b_ k \oplus a_1 b_{k-1} \oplus \cdots \oplus a_{k-1} b_1 \oplus a_ k b_0 \right) \bmod {10}, \]

where any $a_ i$ or $b_ j$ is considered zero if $i > m$ or $j > n$. For example, $9 \otimes 1234$ is $9876$, $90 \otimes 1234$ is $98760$, and $99 \otimes 1234$ is $97536$.

Given $N$, find the smallest positive integer $a$ such that $a \otimes a = N$.

-----Input-----
The input consists of a single line with a positive integer $N$, with at most $25$ digits and no leading zeros.

-----Output-----
Print, on a single line, the least positive number $a$ such that $a \otimes a = N$. If there is no such $a$, print ‘-1’ instead.

-----Examples-----
Sample Input 1:
6
Sample Output 1:
4

Sample Input 2:
149
Sample Output 2:
17

Sample Input 3:
123476544
Sample Output 3:
11112
user: We all know how to alphabetize a list of distinct words when you know the alphabet: One word may be a prefix of another longer word, in which case the shorter word always comes before the longer word. With any other two words there must be a first place in the words where their letters differ. Then the order of the words is determined by the lexicographical order of these first differing letters.

How about the reverse problem: Can you find the lexicographic order of the alphabet from an ordered list of words? Suppose an alphabet exists where the following list of word strings is given in lexicographical order:

cab
cda
ccc
badca

It is clear that c comes before b in the underlying alphabet because cab comes before badca. Similarly, we know a comes before d, because cab < cda, a comes before c because cab < ccc, and d comes before c because cda < ccc. The only ordering of the 4 alphabet characters that is possible is adcb.

However, it may be that a list contains inconsistencies that make it impossible to be ordered under any proposed alphabet. For example, in the following list it must be that a comes before b in the alphabet since abc < bca, yet it also must be that b comes before a in the alphabet since bca < aca.

abc
bca
cab
aca

Finally, some lists may not provide enough clues to derive a unique alphabet order, such as the following:

dea
cfb

In this list, d comes before c but we don’t know about the relative positions of any of the other letters, so we are unable to uniquely discern the order of the alphabet characters.

-----Input-----
The first line of input will contain $L$ and $N$, separated by a space, where $L$ is a lowercase character $\texttt{b} \le L \le \texttt{z}$ representing the highest character in the English alphabet that appears in the derived alphabet, and $N$ is an integer $1 \leq N \leq 1000$ that is equal to the number of strings in the list. Each of the next $N$ lines will contain a single nonempty string of length at most $1000$, consisting only of characters in the derived alphabet. No two strings will be the same.

-----Output-----
If the input is consistent with a unique ordering of the alphabet, output a string that designates that ordered alphabet. If the data is inconsistent with any ordering, output IMPOSSIBLE. If the data is consistent with multiple orderings, output AMBIGUOUS.

-----Examples-----
Sample Input 1:
d 4
cab
cda
ccc
badca
Sample Output 1:
adcb

Sample Input 2:
c 4
abc
bca
cab
aca
Sample Output 2:
IMPOSSIBLE

Sample Input 3:
f 2
dea
cfb
Sample Output 3:
AMBIGUOUS
user: Is a world without cookies really worth living in? Alf and Beata do not think so. That’s why they spent the entire weekend baking an enormous amount of cookies. They baked $A$ almond biscuits, $B$ butter cookies and $C$ chocolate macarons.

It is now time to eat all these delicious cookies with their friends. The cookies will be divided up among all the friends. However, this division must be done such that everyone can eat all of their cookies without having to eat the same kind of cookie twice in a row (that would be very unpleasant).

Furthermore, Alf and Beata wants as few cookies as possible to go to waste – they spent a lot of time preparing them, after all. Can you determine the maximum amount of cookies that can be distributed among everyone?

-----Input-----
The first and only line contains four integers $N$ ($2 \le N \le 10^9$), the number of friends eating cookies including Alf and Beata, and $A$, $B$ and $C$ ($0 \le A, B, C \le 10^9$), the amount of cookies of each of the three times Alf and Beata baked.

-----Output-----
Output a single integer – the number of cookies Alf and Beata are able to distribute.

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

Sample Input 2:
10 20 0 0
Sample Output 2:
10

Sample Input 3:
100 20 543 12
Sample Output 3:
164
user: On his birthday, John’s parents made him a huge birthday cake! Everyone had a wonderful dinner, and now it’s time to eat the cake. There are $n$ candles on the cake. John wants to divide the cake into $n$ pieces so that each piece has exactly one candle on it, and there are no left-over pieces. For that, he made $m$ cuts across the cake. Could you help check if John’s cuts successfully divide the candles on the cake?

Formally, the cake is a circle of radius $r$ centered at $(0, 0)$. The candles are $n$ distinct points located strictly inside the circle. Each cut is a straight line $ax + by + c = 0$, described by three coefficients $a$, $b$, and $c$.

-----Input-----
Input starts with three integers $n$ ($1 \leq n \leq 50$), $m$ ($1 \leq m \leq 15$), and $r$ ($1 \leq r \leq 100$) on the first line.

The next $n$ lines give the locations of the candles. Each line has two integers $x$ and $y$ giving the coordinates of one candle ($0 \leq \sqrt {x^2 + y^2} < r$).

The next $m$ lines give the coefficients of the cutting lines. Each line has three integers $a$, $b$, and $c$ ($0 \leq |a|, |b| \leq 100, 0 \leq |c| \leq 20000$) describing a line of the form $ax + by + c = 0$. The values $a$ and $b$ are not both zero.

All candles and lines are distinct. No candle is on a cut line. No line is completely outside or tangent to the cake. The input guarantees that the number of cake pieces remains the same if any cut line is shifted by at most $10^{-4}$ in any direction. The input also guarantees that each candle remains in the interior of the same piece of cake if its position is shifted by at most $10^{-4}$ in any direction.

-----Output-----
Output “yes” if John’s cuts successfully divide the cake so that each piece he obtains has exactly one candle on it. Otherwise, output “no”.

-----Examples-----
Sample Input 1:
4 2 3
0 1
1 0
-1 0
0 -1
-1 1 0
2 1 0
Sample Output 1:
yes

Sample Input 2:
4 3 3
0 1
1 2
-1 2
0 -1
-1 1 -2
-1 -1 2
0 -1 0
Sample Output 2:
no

Sample Input 3:
3 2 3
2 1
0 0
-1 -2
1 1 -2
3 6 12
Sample Output 3:
yes