meetforever/GRPO_CL huggingface dataset:
user: A set of frogs have accidentally fallen to the bottom of a large pit. Their only means of escaping the pit is to jump out of it. Each frog $i$ is described by three parameters $(l_ i,w_ i,h_ i)$ where $l_ i$ is its leap capacity, $w_ i$ its weight, and $h_ i$ its height. The leap capacity specifies how high that frog can jump. If a frog’s leap capacity is strictly larger than the depth of the pit, the frog can directly escape the pit. However, these frogs are altruistic. Rather than selfishly saving themselves and leaving the frogs with too limited leap capacity behind, they collectively aim to save as many of them from the pit as possible. 

The frogs realize that if a frog $A$ climbs up on the back of frog $B$ before it jumps, the first frog $A$ stands a better chance of escaping the pit: it can escape if $h_ B+l_ A$ is strictly larger than the depth of the pit.

Furthermore, if frog $B$ carrying frog $A$ on its back climbs up on the back of frog $C$, the situation is even better for frog $A$: it can now escape the pit if $h_ C+h_ B+l_ A$ is strictly larger than the depth of the pit.

The frogs can build even higher piles of frogs this way, the only restriction is that no frog may carry other frogs of weight in total amounting to its own weight or heavier. Once a pile has been used to allow a frog to escape, the frogs in the pile jump back to the bottom of the pit and they can then form a new pile (possibly consisting of a different set of frogs). The question is simply how many frogs can escape the pit assuming they collaborate to maximize this number?

-----Input-----
The first line of input contains two integers $n$ and $d$ ($1 \le n \leq 100000$, $1 \le d \le 10^8$), where $n$ is the number of frogs and $d$ is the depth of the pit in µm. Then follow $n$ lines each containing three integers $l, w, h$ ($1 \le l, w, h \le 10^8$), representing a frog with leap capacity $l$ µm, weight $w$ µg, and height $h$ µm. The sum of all frogs’ weights is at most $10^8$ µg.

-----Output-----
Output the maximum number of frogs that can escape the pit.

-----Examples-----
Sample Input:
3 19
15 5 3
12 4 4
20 10 5
Sample Output:
3
user: A Bipartite Graph is an undirected graph whose vertices can be partitioned into $2$ sets such that, for each edge $(u, v)$, $u$ and $v$ belong to different sets.

Socket has challenged Bash to a Bipartite Battle. In the Bipartite Battle, Bash and Socket play a game with Bipartite Graphs.

The Bipartite Battle happens as follows:
 - Socket draws $N$ bipartite graphs. The $i$-th graph has $2$ sets of vertices, one with $a_ i$ vertices, and the other with $b_ i$ vertices.
 - Then the $2$ players, Bash and Socket alternatively take turns. In each turn, a player must choose exactly one non-empty graph, then delete exactly one edge or exactly one vertex of the chosen graph. If the player deletes one vertex, all edges adjacent to it are also deleted.
 - The player who cannot move loses. Note that this can only happen when all edges and all vertices of all graphs have been deleted.
 - Bash plays first.

Of course, Socket does not want to play fair. Socket plans to draw bipartite graphs such that he always wins.

How many ways can Socket draw $N$ bipartite graphs, so that he always wins, assuming both players play optimally?

-----Notes-----
For the $i$-th bipartite graph, let’s number the vertices in first set from $1$ to $a_ i$, and the vertices in second set from $1$ to $b_ i$.

An edge connecting vertex $u$ in first set and vertex $v$ in second set is denoted as $(u, v)$.

Two drawings of bipartite graphs are considered different iff there exists an index $j$ and a pair of integers $(u, v)$, such that:
 - $(u, v)$ is an edge of $j$-th graph in one drawing.
 - $(u, v)$ is NOT an edge of $j$-th graph in the other drawing.

-----Input-----
The first line of input contains the integer $N$ $(1 \le N \le 10^5)$.

$N$ lines follow, each line contains exactly $2$ space-separated integers $a_ i$ and $b_ i$$(1 \le a_ i, b_ i \le 10^9)$.

-----Output-----
Print exactly one integer — the number of ways Socket can draw $N$ bipartite graphs, modulo $10^9 + 7$.

-----Examples-----
Sample Input:
1
1 1
Sample Output:
1
user: You are given a puzzle that can be represented as a $4 \times 4$ grid of colored cells. The solved puzzle contains 4 monochromatic rows, in this order: red, green, blue, yellow. Although we will analyze this puzzle using its 2D representation, it is actually a 3D puzzle! Imagine that the grid is stretched over a torus (in other words, top edge is connected to the bottom one and left edge is connected to the right one). If you are not familiar with the word “torus” or what it is supposed to represent, just replace it with the word(s) “donut (with the hole in the middle)”.For each move you are allowed to either move one row left or right, or one column up or down. The fact that the outer edges are connected means that if a cell is “pushed out” of the grid, it will reappear on the other side of the grid. If you had a torus or a donut handy (or a cup! HAHAha...ha... <sniff>), this would be much clearer. 

Given a description of a state of this puzzle, what is the minimum number of moves you need to solve it? Note that all possible puzzle configurations are solvable in less than 13 moves.

-----Input-----
Input file contains exactly $4$ lines, containing $4$ characters each, each character being either “R”, “G”, “B” or “Y’. The input will describe a valid state of the puzzle.

-----Output-----
Output the minimum number of moves needed to solve the given puzzle.

-----Examples-----
Sample Input:
RGGR
GBGB
BYBY
YRYR
Sample Output:
3
user: Alf and Beata were two kids who lived long ago, during the time before you could spend Halloween participating in programming competitions. As such, their lives were much more boring than the lives of kids today. How can you survive without programming competitions, you may ask yourself. The answer is easy: you go trick-or-treating!

Every year Alf and Beata went out in their neighbourhood to go trick-or-treating. When they trick-or-treat a house, they get some amount of candy of various sorts. Once the kids have filled up their big bag of candy, they go back home.

Since Alf and Beata are very nice siblings, they want to split all their candy evenly. The siblings have $N$ pieces of candy together. However, they each like every piece a bit differently. Alf hates liquorice, which Beata loves. On the other hand, Alf craves chocolate which Beata finds disgusting. More specifically, Alf assigns the $i$’th piece value $a_ i$ while Beata assigns it value $b_ i$. These values can be negative. They want the absolute difference between the sum of the value of all candy each sibling gets to be as small as possible. Note that each candy must be given to one of the siblings: they can’t throw any candy away.

-----Input-----
The first line of the input contains a single integer $N$ ($1 \le N \le 100$), the number of candies the siblings have collected. The next line contains the $N$ integers $a_1, a_2 \dots , a_ N$ ($-100 \le a_ i \le 100$). The third and final line contains the $N$ integers $b_1, b_2 \dots , b_ N$ ($-100 \le b_ i \le 100$).

-----Output-----
Output a string containing $N$ characters. The $i$’th character should be A if the candy should be given to Alf, and B if it should be given to Beata. If there are multiple solutions, output the lexicographically smallest one.

-----Examples-----
Sample Input:
5
-2 -1 0 1 2
2 1 0 -1 -2
Sample Output:
AAAAA
user: When constructing electric circuits one has to connect pairs of points using wire, preferable as short as possible. In this problem we have an empty circuit board of size $N \times M$ where we want to connect the two points $A_1$ and $A_2$ with each other using one wire, and the two points $B_1$ and $B_2$ with each other using another wire. The wires must go along the horizontal and vertical edges of the grid (see figure), and the two wires may not share a common vertex. Determine the minimum length of wire needed to do so. The wire may not go outside the circuit board.

-----Input-----
The first line contains two integers, $N$ ($2 \le N \le 100$) and $M$ ($2 \le M \le 100$), the grid size of the circuit board.

Then follows four lines containing the coordinates for the points $A_1$, $A_2$, $B_1$ and $B_2$, respectively. Each coordinate pair will be described using two integers and will correspond to an intersection point in the grid. The first coordinate will be between $0$ and $N$ inclusive and the second coordinate between $0$ and $M$ inclusive. All coordinate pairs will be unique.

-----Output-----
A single line containing the minimum length of wire needed to connect the points, or “IMPOSSIBLE” if it’s not possible to do so.

-----Examples-----
Sample Input:
6 3
2 3
4 0
0 2
6 1
Sample Output:
IMPOSSIBLE
user: Typing on phones can be tedious. It is easy to make typing mistakes, which is why most phones come with an autocorrect feature. Autocorrect not only fixes common typos, but also suggests how to finish the word while you type it. Jenny has recently been pondering how she can use this feature to her advantage, so that she can send a particular message with the minimum amount of typing. 

The autocorrect feature on Jenny’s phone works like this: the phone has an internal dictionary of words sorted by their frequency in the English language. Whenever a word is being typed, autocorrect suggests the most common word (if any) starting with all the letters typed so far. By pressing tab, the word being typed is completed with the autocorrect suggestion. Autocorrect can only be used after the first character of a word has been typed – it is not possible to press tab before having typed anything. If no dictionary word starts with the letters typed so far, pressing tab has no effect.

Jenny has recently noticed that it is sometimes possible to use autocorrect to her advantage even when it is not suggesting the correct word, by deleting the end of the autocorrected word. For instance, to type the word “autocorrelation”, Jenny starts typing “aut”, which then autocorrects to “autocorrect” (because it is such a common word these days!) when pressing tab. By deleting the last two characters (“ct”) and then typing the six letters “lation”, the whole word can be typed using only $3$ (“aut”) ${}+1$ (tab) ${}+2$ (backspace twice) ${}+6$ (“lation”) $= 12$ keystrokes, $3$ fewer than typing “autocorrelation” without using autocorrect.

Given the dictionary on the phone and the words Jenny wants to type, output the minimum number of keystrokes required to type each word. The only keys Jenny can use are the letter keys, tab and backspace.

-----Input-----
The first line of input contains two positive integers $n$ ($1 \le n \le 10^5$), the number of words in the dictionary, and $m$ ($1 \le m \le 10^5$), the number of words to type. Then follow $n$ lines with one word per line, sorted in decreasing order of how common the word is (the first word is the most common). No word appears twice in the dictionary. Then follow $m$ lines, containing the words to type.

The dictionary and the words to type only use lower case letters ‘a’-‘z’. The total size of the input file is at most $1$ MB.

-----Output-----
For each word to type, output a line containing the minimum number of keystrokes required to type the corresponding word.

-----Examples-----
Sample Input:
5 5
austria
autocorrect
program
programming
computer
autocorrelation
programming
competition
zyx
austria
Sample Output:
12
4
11
3
2
user: People are going to the movies in groups (or alone), but normally only care to socialize within that group. Being Scandinavian, each group of people would like to sit at least one space apart from any other group of people to ensure their privacy, unless of course they sit at the end of a row. The number of seats per row in the cinema starts at $X$ and decreases with one seat per row (down to a number of $1$ seat per row). The number of groups of varying sizes is given as a vector $(N_1, \ldots , N_ n)$, where $N_1$ is the number of people going alone, $N_2$ is the number of people going as a pair etc. Calculate the seat-width, $X$, of the widest row, which will create a solution that seats all (groups of) visitors using as few rows of seats as possible. The cinema also has a limited capacity, so the widest row may not exceed $12$ seats.

-----Input-----
The first line of input contains a single integer $n$ ($1 \leq n \leq 12$), giving the size of the largest group in the test case.

Then follows a line with $n$ non-negative integers, the $i$-th integer ($1$-indexed) denoting the number of groups of $i$ persons who need to be seated. Each such number is at most $30$, and not all of them are $0$.

-----Output-----
A single number; the size of the smallest widest row that will accommodate all the guests. If this number is greater than $12$, output impossible instead.

-----Examples-----
Sample Input:
3
0 1 1
Sample Output:
3
user: David is writing an article for the Bulletin of the Association of Proof Completions. In his article, he proves several theorems. For every theorem, David came up with a proof. Since David is a very eager student, he even came up with multiple proofs for some of the theorems. As usual, a proof for a theorem may depend on a number of other theorems.

The article has to be as short as possible to publish it, and David only really cares about the main theorem, Theorem $0$. In order to achieve this, he has estimated the number of words he will need for every proof. Can you help David find the shortest possible length of his article?

-----Input-----
 - A single line containing $1\leq n\leq 20$, the number of theorems.
 - For each theorem:
 - A single line containing $1\leq p_ i\leq 10$, the number of proofs for the $i$th theorem.
 - $p_ i$ lines, each of the form $l$, $k$, $d_0,\dots ,d_{k-1}$, where $0\leq l\leq 10^6$ is the length of the proof, $0\leq k\leq n-1$ is the number of theorems the proof depends on, and the $0\leq d_ i\leq n-1$ are the numbers of the theorems the proof depends on.

-----Output-----
Print one line with a single integer, the shortest possible length of David’s article.

-----Examples-----
Sample Input:
2
2
10 0
3 1 1
1
4 1 0
Sample Output:
10
user: A dishonest landowner is selling off plots of land. He’s selling it in large, rectangular plots, but many of the plots overlap, so he’s actually selling the same land multiple times! It’s not real estate, it’s unreal estate!

Given a description of the possibly overlapping rectangular plots of land that the dishonest landowner has sold, determine the total actual area of land covered by them.

-----Input-----
There will be a single test case in the input. Each test case will begin with a line with a single integer $n$ ($0 < n \le 5000$), indicating the number of plots of land sold. The next $n$ lines will each have a description of a rectangular plot of land, consisting of four real numbers with at most two decimals after the decimal point:

     x1 y1 x2 y2

where $(x_1,y_1)$ is the southwest corner, and $(x_2,y_2)$ is the northeast corner ($-1000 \le x_1 < x_2 \le 1000$, $-1000 \le y_1 < y_2 \le 1000$).

-----Output-----
Output a single real number, which represents the total actual area covered by all of the rectangular plots of land. Output this number with exactly two decimal places, rounded.

-----Examples-----
Sample Input:
2
0 0 100 100
30 30 60 60
Sample Output:
10000.00
user: Pia is a famous bartender at the hip Stockholm night club Supernova. One of her most impressive feats is the mixing a series of drinks using each of the $N$ distinct drink ingredients in the bar exactly once. She does this in the following way.

First, Pia chooses a number of drinks to make. Each of the drink ingredients are then lined up in front of her in order $1, 2, \dots , N$. For the first drink, she uses some positive number $K$ of ingredients starting from the left, i.e. $1, 2, ..., K$. For the next drink, she uses some positive number $L$ of ingredients starting from the first unused ingredient, i.e. $K + 1, K + 2, \dots , K + L$. She continues this process until the final drink, which uses some set of ingredients $N - M, N - M + 1, \dots , N$.

However, not every pair of ingredients work well in a drink. For example, milk and water would not go very well together. She may not include a bad pair of ingredients in any drink.

So far, she has managed to make a different set of drinks every night. For how many nights can she mix a new set of drinks? We call two sets of drinks different if they do not consist of the exact same drinks (though they are allowed to have drinks in common).

-----Input-----
The first line of the input contains two integers $1 \le N \le 100000$ and $0 \le P \le 100000$, the number of ingredients and bad pairs of ingredients.

Each of the next $P$ lines contains two integers $1 \le a \not= b \le N$, two ingredients that do not work well together in a drink. The same pair of ingredients may appear multiple times in this list.

-----Output-----
Output a single integer, the number of nights Pia can construct a different set of drinks. Since this number may be large, output the remainder when divided by $10^9 + 7$.

-----Examples-----
Sample Input:
5 3
1 3
4 5
2 4
Sample Output:
5
user: After your boat ran out of fuel in the middle of the ocean, you have been following the currents for 80 days. Today, you finally got your radar equipment working. And it’s receiving signals!

Alas, the signals come from the “radar” station owned by the eccentric lighthouse keeper Hasse. Hasse’s radar station (which does not work quite like other radar stations) emits continuous signals of three different wave-lengths. Therefore, the only interesting thing you can measure is the phase of a signal as it reaches you. For example, if the signal you tuned on to has a wave-length of $100$ meters and you are $1456$ meters from the station, your equipment can only tell you that you are either $56$, or $156$, or $256$, or $\dots $ meters away from the lighthouse.

So you reach for your last piece of paper to start calculating – but wait, there’s a catch! On the display you read: “ACCURACY: 3 METERS”. So, in fact, the information you get from this signal is that your distance from Hasse’s radar station is in the union of intervals $[53,59] \cup [153, 159] \cup [253, 259] \cup \dots $.

What to do? Since the key to surviving at sea is to be optimistic, you are interested in what the smallest possible distance to the lighthouse could be, given the wavelengths, measurements and accuracies corresponding to the three signals.

-----Task-----
Given three positive prime numbers $m_1$, $m_2$, $m_3$ (the wavelengths), three nonnegative integers $x_1$, $x_2$, $x_3$ (the measurements), and three nonnegative integers $y_1$, $y_2$, $y_3$ (the accuracies), find the smallest nonnegative integer $z$ (the smallest possible distance) such that $z$ is within distance $y_ i$ from $x_ i$ modulo $m_ i$ for each $i = 1,2,3$. An integer $x’$ is within distance$y$ from $x$ modulo $m$ if there is some integer $t$ such that $x \equiv x’ + t \pmod{m}$ and $|t| \leq y$.

-----Input-----
There are three lines of input. The first line is $m_1$ $m_2$ $m_3$, the second is $x_1$ $x_2$ $x_3$ and the third is $y_1$ $y_2$ $y_3$. You may assume that $0 < m_ i \leq 10^6$, $0 \leq x_ i < m_ i$, and $0 \leq y_ i \leq 300$ for each $i$. The numbers $m_1$, $m_2$, $m_3$ are all primes and distinct.

-----Output-----
Print one line with the answer $z$. Note that the answer might not fit in a 32-bit integer.

-----Examples-----
Sample Input:
11 13 17
5 2 4
0 0 0
Sample Output:
2095
user: Dino loves ice cream. In case he ever run out of ice cream at his office, he keeps a stash of dry ice in order to quickly make new ice cream.

His recipe for making ice cream includes exactly $T$ liters of dry ice. Unfortunately, he has no marked containers in his office. Instead, he keeps a set of bottles of known total volume.

He wants to use this in order to transfer $T$ liters of dry ice from his dry ice container to the container in which he is mixing his ice cream. To do this, he should perform a sequence of three different kinds of actions. He may either fill a bottle with dry ice from the dry ice container until the bottle is full, empty the contents of a bottle into the ice container, or transfer dry ice from one bottle into other until either the first bottle becomes empty or the target bottle becomes full.

Can you help Dino construct a plan in order to transfer $T$ liters of dry ice into the ice cream mix?

-----Input-----
The first line of the input contains an integer $1 \le N \le 100$, the number of bottles.

The next line contains $N$ integers, separated by spaces. These are the volumes of all the bottles, in liters. Each volume is between $1$ and $100$ liters.

The final line contains the integer $1 \le T \le 100$, the volume in liters of dry ice needed for the ice cream.

-----Output-----
If it is not possible to add the correct amount of dry ice, output impossible. Otherwise, output a sequence of moves that moves $T$ liters into the ice cream mix.

You may output the following moves:
 - fill x: Fill bottle $x$ from the ice cream container until it is full.
 - discard x: Empty bottle $x$ into the sink.
 - transfer x y: Pour from bottle $x$ into bottle $y$ until either $y$ is full or $x$ is empty.

When pouring dry ice into the ice cream mix, use $y = 0$ as the target bottle.

You may output at most $100000$ moves.

-----Examples-----
Sample Input:
2
7 8
10
Sample Output:
fill 2
transfer 2 1
transfer 2 0
discard 1
fill 2
transfer 2 1
transfer 2 0
discard 1
fill 2
transfer 2 0
user: The enemy and their massive army is approaching your fortress, and all you have to defend it is a legion of guardian gnomes. There is no hope of winning the battle, so your focus will instead be on causing as much damage as possible to the enemy. 

You have $n$ gnomes at your disposal. Before the battle, they must be divided into at most $m$ non-empty groups. The battle will then proceed in turns. Each turn, your gnomes will attack the enemy, causing one unit of damage for each living gnome. Then the enemy will attack by throwing a lightning bolt at one of the $m$ groups. The lightning bolt kills $k$ of the gnomes in that group, or all of them if the number of living gnomes in the group is less than $k$. The battle ends when all gnomes are dead. The enemy will always throw the lightning bolts in an optimal way such that the total damage caused by the gnomes is minimized.

Now you wonder, what is the maximum amount of damage you can cause to the enemy if you divide the gnomes into groups in an optimal way?

For example, if as in Sample Input 1 you have $n=10$ gnomes that are to be divided into $m=4$ groups, and the lightning bolt does at most $k=3$ damage, then an optimal solution would be to create one large group of size $7$ and three small groups of size $1$. In the first round, you cause $10$ damage and the lightning bolt reduces the large group by $3$. In the next round, you cause $7$ damage and the large group is reduced down to size $1$. In the remaining four rounds you do $4$, $3$, $2$, and $1$ damage respectively and the lightning bolt removes one group each round. In total you do $10+7+4+3+2+1 = 27$ damage.

-----Input-----
The input consists of a single line containing the three integers $n$, $m$, and $k$ ($1 \le n \le 10^9$, $1 \le m, k \le 10^7$), with meanings as described above.

-----Output-----
Output the maximum amount of damage you can cause to the enemy.

-----Examples-----
Sample Input:
10 4 3
Sample Output:
27