meetforever/GRPO_CL huggingface dataset:
user: Peter is in charge of food service on a passenger train. The train has $n$ cars ordered sequentially starting from the locomotive. The $i$-th car has $p_ i$ passengers. There are $m$ food carts Peter may put into service. Each food cart serves a unique type of food. The $j$-th food cart can move between the $l_ j$-th car and the $r_ j$-th car (both inclusive) and serve the passengers in these cars.

The passengers on the train are not only hungry, but also adventurous: as long as a food cart is in service, they would love to try it! A passenger will be happy if he/she sits in a car that is served by all the food carts that are in service. Peter would like to design service plans that run one or more of the $m$ food carts so that at least $k$ of the passengers on the train are happy.

Peter wants to count the number of different service plans he may choose from (modulo $10^9 + 7$). Two service plans are different if one food cart is put into service in one plan but not in the other plan.

-----Input-----
The first line has three integers $n$, $m$, and $k$ ($1 \leq n, m \leq 2 \cdot 10^5, 1 \leq k \leq 10^{14}$). The next line has $n$ integers. The $i$-th integer is $p_ i$ ($1 \leq p_ i \leq 10^9 $), the number of passengers in the $i$-th car. Each of the next $m$ lines describes a food cart. The $j$-th of these lines has two integers $l_ j$ and $r_ j$ ($1 \leq l_ j \leq r_ j \leq n$), giving the service range of the $j$-th food cart.

-----Output-----
Output the number of different service plans, modulo $10^9 + 7$.

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

Sample Input 2:
5 4 6
1 2 3 4 5
1 4
2 3
3 5
4 5
Sample Output 2:
5
user: Consider a very large number $R$ in a compressed format. It is compressed as a binary string $s$, and an integer $k$. Start with the empty string, and append $s$ to it $k$ times to get the binary representation of $R$. The string $s$ is guaranteed to have a leading $1$. Now, with $R$, solve the following problem: How many sets of $n$ distinct integers are there such that each integer is between 0 and $R-1$, inclusive, and the XOR of all those integers is equal to zero? Since this number can get very large, return it modulo $10^{9}+7$.

As a reminder, XOR is Exclusive Or. The XOR of two numbers is done bitwise. Using $\oplus $ for XOR:\[ 0\oplus 0=0 \]\[ 0\oplus 1=1 \]\[ 1\oplus 0=1 \]\[ 1\oplus 1=0 \]

XOR is associative, so $a\oplus (b\oplus c) = (a\oplus b)\oplus c$.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each input consists of exactly two lines. The first line has two space-separated integers $n$ ($3\le n \le 7$) and $k$ ($1 \le k \le 100000$), where $n$ is the number of distinct integers in the sets, and $k$ is the number of times to repeat the string $s$ in order to build $R$. The second line contains only the string $s$, which will consist of at least 1 and at most 50 characters, each of which is either 0 or 1. $s$ is guaranteed to begin with a 1.

-----Output-----
Output a single line with a single integer, indicating the number of sets of $n$ distinct integers that can be formed, where each integer is between 0 and $R-1$ inclusive, and the XOR of the $n$ integers in each set is 0. Output this number modulo $10^9+7$.

-----Examples-----
Sample Input 1:
3 1
100
Sample Output 1:
1

Sample Input 2:
4 3
10
Sample Output 2:
1978
user: Luke likes to ride on public transit in different cities he visits, just for fun. He tries to find unique ways to travel in loops: leaving from one transit station, traveling along the transit connections to at least one other station, and returning to the station where he started. He is finding lots of loops, and he wants to know just how many there are in different transit systems. There may be so many he won’t ever have time to try them all, but he’ll find some satisfaction in knowing they are there.

He’s particularly interested in counting simple loops. A simple loop is a sequence of unique transit stations $t_1, t_2, \ldots , t_ j$, where there’s a way to connect directly from $t_ i$ to $t_{i+1}$ for $1 \leq i < j$ and also from $t_ j$ to $t_1$. Of course, we can write down a simple loop starting with of the stations in the loop, therefore we consider any cyclic shift of such a sequence to be the same simple loop. However, two simple loops which visit the same set of transit stations in a different order are considered distinct.

Help Luke by writing a program to count how many unique simple loops there are in each transit system. The following figures illustrate the transit stations (numbered ovals) and one-way connections (arrows) of the sample input.  

-----Input-----
Input contains a description of one transit system. The description begins with a line containing an integer $3 \leq m \leq 9$ indicating the number of transit stations in the system. Stations are numbered $0$ to $m-1$. The next line contains an integer $1 \leq n \leq m(m-1)$ indicating the number of connections that follow, one connection per line. Each connection is a pair of integers $s~ t$ ($0 \leq s < m$, $0 \leq t < m$, $s \neq t$), indicating that there is a one-way connection from station $s$ to station $t$.

-----Output-----
Print the number of unique simple loops in the transit system.

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

Sample Input 2:
8
10
0 1
1 2
2 3
3 4
4 5
5 0
2 6
6 0
3 7
7 0
Sample Output 2:
3
user: You are given a rooted tree with $n$ nodes. The nodes are numbered $1..n$. The root is node $1$, and $m$ of the nodes are colored red, the rest are black.

You would like to choose a subset of nodes such that there is no node in your subset which is an ancestor of any other node in your subset. For example, if A is the parent of B and B is the parent of C, then you could have at most one of A, B or C in your subset. In addition, you would like exactly $k$ of your chosen nodes to be red.

If exactly $m$ of the nodes are red, then for all $k=0..m$, figure out how many ways you can choose subsets with $k$ red nodes, and no node is an ancestor of any other node.

-----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 begin with a line with two integers $n$ ($1 \le n \le 2 \times 10^5$) and $m$ ($0 \le m \le min(10^3,\ n)$), where $n$ is the number of nodes in the tree, and $m$ is the number of nodes which are red. The nodes are numbered $1..n$.

Each of the next $n-1$ lines will contain a single integer $p$ ($1 \le p \le n$), which is the number of the parent of this node. The nodes are listed in order, starting with node $2$, then node $3$, and so on. Node $1$ is skipped, since it is the root. It is guaranteed that the nodes form a single tree, with a single root at node $1$ and no cycles.

Each of the next $m$ lines will contain single integer $r$ ($1 \le r \le n$). These are the numbers of the red nodes. No value of $r$ will be repeated.

-----Output-----
Output $m+1$ lines, corresponding to the number of subsets satisfying the given criteria with a number of red nodes equal to $k=0..m$, in that order. Output this number modulo $10^9+7$.

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

Sample Input 2:
4 4
1
1
1
1
2
3
4
Sample Output 2:
1
4
3
1
0
user: When ninjas go to conferences they wear fake nametags. One ninja in particular wants to impress his Sensei. His Sensei chooses a new favorite number every day. The pupil wants to put a name on his nametag that encodes his Sensei’s favorite number! This name will consist of only lower case letters. He assigns a value to each letter, based on its position in the alphabet (e.g. $\texttt{a}=1$, $\texttt{b}=2$, ..., $\texttt{z}=26$). Then, he encodes the Sensei’s number by adding up the absolute values of the differences of every consecutive pair of letters. For example, the string $\texttt{azxb}$ has the value of:\[ |\texttt{a} - \texttt{z}| + |\texttt{z} - \texttt{x}| + |\texttt{x} - \texttt{b}| = |1 - 26| + |26 - 24| + |24 - 2| = 49 \]

The name that the ninja will write on his nametag is the shortest string that encodes to his Sensei’s favorite number. If there’s more than one string of the shortest length, he’ll choose the one that comes first alphabetically. Given the Sensei’s favorite number, $k$, find the string that the ninja should put on his nametag.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each input consists of a single line with a single integer $k$ ($1 \le k \le 1000000$), which is the Sensei’s favorite number. There will always be a name that encodes to the Sensei’s number.

-----Output-----
Output a single line with a string of lower case letters, which is the name that the ninja should put on the nametag to impress the Sensei.

-----Examples-----
Sample Input 1:
1
Sample Output 1:
ab

Sample Input 2:
19
Sample Output 2:
at
user: A haiku is a Japanese form of poetry. It consists of three phrases of $5$, $7$ and $5$ syllables each.

Once a year, HiQ has a haiku contest, where employees submit their best poems. The poems are then judged based on a wide range of aspects, such as
 - creativity
 - simplicity
 - beauty
 - whether the poem is actually a haiku

This last point turned out to be quite a challenge for the judges (many problems arose when half the judges indexed syllables starting at $0$ and the other half at $1$).

Can you help the judges to determine whether a submitted poem is a haiku, given a set of syllables? Note that there may exist multiple decompositions of a single word in the poem into syllables. In this case, you should determine whether some decomposition is a haiku.

-----Input-----
The first line of input contains a single integer $1 \le S \le 100$, the number of syllables. The next line contains $S$ words separated by spaces (the syllables). Each syllable contains at most $7$ lowercase letters a-z.

Then, three lines containing the poem follow. Each line contains a (non-empty) list of words separated by spaces, representing a phrase. The words contain only lowercase letters a-z. The length of each line is at most $100$ characters (including spaces).

It is guaranteed that there exists at least one decomposition of the poem into the given syllables.

-----Output-----
Output “haiku” if the given poem is a haiku, and “come back next year” otherwise (without the quotes).

-----Explanation of Sample Input 1-----
One possible decomposition into a haiku is:

spe-lling ve-ry hard
ear-ly in mor-ning ti-red
i need cov-fe-fe


-----Explanation of Sample Input 3-----
No matter how the third line is decomposed, it contains $8$ syllables instead of $5$, so it can not be a haiku.

-----Examples-----
Sample Input 1:
20
va
fi
ve
mor
lling
spe
in
i
sh
ti
hard
cov
ear
fe
red
ja
ry
need
ly
ning
spelling very hard
early in morning tired
i need covfefe
Sample Output 1:
haiku

Sample Input 2:
22
q
c
da
plus
to
the
e
thee
un
late
ci
a
shall
by
com
i
ru
pare
temp
ble
hi
de
shall i compare thee
to a c plus plus template
undecidable
Sample Output 2:
haiku
user: Concentration is a not so popular 2 player card game of both skill and luck. The standard Concentration game is played with one or two 52-card decks, however, for the sake of the problem, we will look at a variation of Concentration.

The rules are as follows:
 - A card is represented by a single integer. Two cards $i$, $j$ are considered “similar” if and only if $\lfloor i/2\rfloor =\lfloor j/2\rfloor $. A deck consisting of $2N$ cards is used for each game. More specifically, a deck of $2N$ cards contains exactly one copy of card $i$ for all $0\leq i <2N$.
 - All cards are initially facing down on a table in random positions, i.e. neither players know what any cards are. Players take turns making moves. Player 0 goes first, then player 1 goes, then player 0 goes, and so on.
 - During each turn, a player chooses two cards and reveals them. If the two cards are “similar”, then they are removed from the table and the player gets to keep them, the player is then granted another turn; this can happen infinitely as long as the player always finds two “similar” cards. If the cards are different, the player’s turn ends.
 - When there are no more cards on the table, the player with more cards wins the game.

Anthony and Matthew like to play this boring game and share an identical play style: whenever they are to choose a card to reveal, if they have knowledge of two “similar” cards, they will pick one of the two “similar” cards; otherwise they will pick a random unknown card to reveal.

Anthony and Matthew are both extremely intelligent and have perfect memories, i.e. they remember every card that has been revealed.

Before the game starts, both Anthony and Matthew make up their minds about in which order they will choose random cards to reveal, in case when they do not have knowledge of two “similar” cards.

Each player’s choices of revelation can be represented by a permutation of numbers $[0,\ldots , 2N-1]$. For example, let $\sigma _0$, a permutation of $[0,\ldots , 2N-1]$ be the “random” choices of Anthony. When Anthony is to choose an unknown card, he will choose the smallest $i$ such that $\sigma _0(i)$ is not revealed, and reveal $\sigma _0(i)$.

Similarly, let $\sigma _1$ be the choices of Matthew.

Having knowledge of $\sigma _0$ and $\sigma _1$, we should be able to perfectly determine the winner (and win lots of money by betting on that player), and it is your job to do exactly that!

-----Input-----
The first line of input contains one integer $1\leq N\leq 10^6$.

The second line contains $2N$ integers, with the $i$-th integer being $\sigma _0(i)$. This line defines $\sigma _0$.

The third line contains $2N$ integers, with the $i$-th integer being $\sigma _1(i)$. This line defines $\sigma _1$.

-----Output-----
Output a single line with $0$ if Anthony wins, $1$ if Matthew wins, or $-1$ if the game ties.

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

Sample Input 2:
2
0 2 1 3 
0 2 1 3 
Sample Output 2:
1
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 $e$ ($0 \le n \le 15000000, 0 \le e \le 25$).

-----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
user: Vjekoslav the Wolf is running away from a bunch of blood hungry hunters. The hunters are smart and hide behind trees. Vjekoslav knows this, but doesn’t know which trees. He would like to run to his comfortable, civilized cottage (as opposed to the hunters quite uncivilized den, yes I am rooting for the Wolf here) staying as far away as possible from any trees.

The forest can be represented as an $N$ by $M$ grid. Let us mark empty meadow patches with ‘.’, patches with a tree in the middle with ‘+’, Vjekoslav’s current position with ‘V’ and the position of his cottage with ‘J’. Vjekoslav can run from his current patch to any other patch north, east, south or west from him, even if it contains a tree.

If Vjekoslav is standing in $R$-th row and $C$-th column on the grid and there is a tree in the $A$-th row and $B$-th column then the distance between Vjekoslav and that tree is\[ \lvert R-A \rvert + \lvert C-B \rvert \]

Help Vjekoslav find the best route to his cottage. The best route is any route that maximizes the minimal distance between Vjekoslav and all trees at any given moment.

Note that Vjekoslav’s cottage doesn’t occupy the entire patch so that patch must also be included in the route.

-----Input-----
The first line of input contains integers $N$ and $M$ ($1 \leq N, M \leq 500$), the grid dimensions.

The next $N$ lines contain $M$ characters, each being one of ‘.’, ‘+’, ‘V’, and ‘J’.

The input will contain exactly one each of the characters ‘V’ and ‘J’, and at least one character ‘+’.

-----Output-----
Output a single integer, the minimal distance from a tree in the optimal route.

-----Examples-----
Sample Input 1:
4 4
+...
....
....
V..J
Sample Output 1:
3

Sample Input 2:
4 5
.....
.+++.
.+.+.
V+.J+
Sample Output 2:
0
user: In competitive weightlifting, you must perform a sequence of lifts. You have a constant strength $s$, and a decreasing energy reserve $e$. For each lift, you may choose any positive (not necessarily integer) weight $w$ to attempt. If $s \ge w$, the lift succeeds and your energy goes down by $e_{\text {success}}$; if $s < w$, the lift fails and your energy goes down by $e_{\text {failure}}$. You may continue attempting lifts as long as $e > 0$. If at any point $e \le 0$, you can make no further attempts. Your score is the maximum weight you successfully lift or $0$ if every attempt failed.

Ideally, you should lift exactly at your strength limit. However, you do not know your strength $s$. You only know that you can definitely lift the empty bar ($25\text { kg}$), and that the maximum conceivable lift is $225\text { kg}$. How close to an optimal score can you guarantee? That is, what’s the smallest $d$ for which you can ensure a score of at least $s-d$?

For example, suppose $e = 4$, $e_{\text {success}} = 1$ and $e_{\text {failure}} = 2$. You try to lift $200\text { kg}$ and fail. Now, $e = 2$. You try $100\text { kg}$ and succeed. Now, $e = 1$. You try $150\text { kg}$ and succeed. Now, $e = 0$ and you must stop. You know that you can lift $150\text { kg}$, but you cannot lift $200\text { kg}$. Your strength $s$ must be somewhere between $150\text { kg}$ and $200\text { kg}$. You scored $150$, your optimal score might be as high as (just under) $200$. You still don’t know $s$, but you know you’re within $50$. In this case, $d = 50$.

That’s a specific example, and the strategy used is certainly not optimal. You can do better. What’s the smallest value of $d$ you can get so that you can guarantee a score of at least $s-d$ for any and all possible values of $s$?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The input consists of a single line with $3$ space-separated integers $e$, $e_{\text {success}}$, $e_{\text {failure}}$ ($1 \le e, e_{\text {success}}, e_{\text {failure}} \le 10^7$) where $e$ is your beginning energy reserve, $e_{\text {success}}$ is the amount of energy expended in a successful lift, and $e_{\text {failure}}$ is the amount of energy expended in a failed lift.

-----Output-----
Output a single line with a real number $d$, which is the minimum weight in $\text {kg}$ such that you can ensure a score of at least $s-d$. Your answer will be considered correct if its absolute or relative error does not exceed $10^{-6}$.

-----Examples-----
Sample Input 1:
1 3 3
Sample Output 1:
112.5

Sample Input 2:
12 3 3
Sample Output 2:
13.333333333333334
user: There are $N$ balloons floating in the air in a large room, lined up from left to right. Young Perica likes to play with arrows and practice his hunting abilities. He shoots an arrow from the left to the right side of the room from an arbitrary height he chooses. The arrow moves from left to right, at a chosen height $H$ until it finds a balloon. The moment when an arrow touches a balloon, the balloon pops and disappears and the arrow continues its way from left to right at a height decreased by $1$. Therefore, if the arrow was moving at height $H$, after popping the balloon it travels on height $H-1$.

Our hero’s goal is to pop all the balloons using as little arrows as possible.

-----Input-----
The first line of input contains the integer $N$ ($1 \leq N \leq 1\ 000\ 000$). The second line of input contains an array of $N$ integers $H_ i$. Each integer $H_ i$ ($1 \leq H_ i \leq 1\ 000\ 000$) is the height at which the $i$-th balloon floats, respectively from left to right.

-----Output-----
The first and only line of output must contain the minimal number of times Pero needs to shoot an arrow so that all balloons are popped.

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

Sample Input 2:
5
1 2 3 4 5
Sample Output 2:
5
user: The boardgame Chaos is an exotic variant of Chess, played by two players in alternating turns on an $n\times n$ playing board. All pieces have the same set of $n$ valid moves which are agreed on ahead of the game. 

In a single turn a player can pick exactly one of their pieces and perform one of the following actions:
 - Perform up to two valid moves using the chosen piece, capturing any piece that the chosen piece lands on along the way.
 - Teleport the chosen piece to any cell on the board that is not already occupied by another piece.
 - Leave the chosen piece untouched in its current cell.

Having recently discovered Chaos, Alice and Bob are currently in the endgame of a very exciting match. Each player has a single piece left on the board and there are only two turns left, with Alice going next.

Having analysed the situation, she realises that the only way she can win is to capture Bob’s piece in her turn. If that is not possible, Alice may be able to force a tie if she can teleport her piece to a cell that Bob cannot capture in his turn. Otherwise Bob will be able to win by capturing Alice’s piece, no matter what she does in her turn. Help Alice determine her optimal outcome.

-----Input-----
The input consists of:
 - One line with an integer $n$ ($2 \leq n \leq 10^5$), the size of the playing board and the number of valid moves.
 - One line with two integers $a_ x$ and $a_ y$ ($1 \leq a_ x, a_ y \leq n$), the column and row in which Alice’s piece is currently located.
 - One line with two integers $b_ x$ and $b_ y$ ($1 \leq b_ x, b_ y \leq n$), the column and row in which Bob’s piece is currently located.
 - $n$ lines, the $i$th of which contains two integers $x_ i$ and $y_ i$ ($-n < x_ i, y_ i < n$) representing one of the valid moves. This moves the given piece $x_ i$ columns to the right and $y_ i$ rows up, provided this does not take the piece outside of the board.

Columns are numbered $1$ to $n$ from left to right and rows are numbered $1$ to $n$ from bottom to top. All valid moves are distinct.

-----Output-----
If Alice can capture Bob’s piece in her turn, output “Alice wins”.

If Alice can use her turn to force a tie by teleporting her piece to a cell that Bob cannot capture in his turn output “tie” followed by two integers $a’_ x$ and $a’_ y$, the location of any such cell. If there are multiple valid solutions, you may output any one of them.

Otherwise, if Bob is able to capture Alice’s piece no matter what she does in her turn, output “Bob wins”.

-----Examples-----
Sample Input 1:
2
2 1
1 2
1 0
0 -1
Sample Output 1:
Bob wins

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