meetforever/GRPO_CL huggingface dataset:
user: You are the boss of ACM (Association for the Country of Mububa), an upstanding company with a single goal of world domination.

Today, you have conquered the unnamed country of Mububa (how an unnamed country has a name is, of course, outside the scope of this problem). Mububa is known for its great, great, bananas. In light of this monumental achievement, you have decided to reward your executives with Mububa’s greatest treasure (which is obviously, bananas). You have prepared $N$ briefcases, each contains a number of bananas. These briefcases are numbered from $1$ through $N$.

You reward your executives one by one in order from the least evil executive, to the most evil executive (still not comparably evil to you, of course). No two executives are equally evil. For each executive, you first decide how many briefcases you want to give him. If you decide to give an executive $a$ briefcases, you give him the $a$ briefcases with lowest numbers that you still have. Each executive you reward must receive at least one briefcase.

It is important to be fair when distributing rewards. You do not want your executives to stage a hunger strike, after all. Thus, the rewards the executives received must reflect how evil they are. More rigorously, if executive $A$ is more evil than executive $B$, then the total number of bananas received by executive $A$ must be at least as large as the total number of bananas received by executive $B$.

You know the number of bananas inside all of the briefcases. You want to reward as many executives as possible, but wants the distribution to still be fair (i.e. following the previous requirement) amongst them. What is the maximum number of executives you can reward this way?

-----Input-----
The first line contains a non-negative integer $2 \leq N \leq 3000$, giving the number of briefcases you have. Then follows a line with $N$ integers, the $i$-th of which denotes the number of bananas in briefcase number $i$. Each briefcase contains between $1$ and $10^9$ bananas, inclusively.

-----Output-----
Print the maximum number of executives you can reward with bananas.Sample Data explanation

In the first example, give briefcase $1$ to the least evil executive, briefcase $2$ to the second least evil executive, and briefcases $3$ and $4$ to the most evil executive.

In the second example, give briefcase $1$ to the least evil executive, briefcases $2$ and $3$ to the second least evil executive, and briefcases $4$, $5$, and $6$ to the most evil executive.

-----Examples-----
Sample Input:
4
1 2 1 2
Sample Output:
3
user: Alice has an $n\! \times \! m$ grid and a $2\! \times \! 2$ block. She would like to place her block in the grid. She must place it so that the block is axis-aligned and covers exactly $4$ grid cells.

Bob wants to prevent Alice from doing that. To do this, he places obstacles in some of the grid cells. After Bob places his obstacles, all $2\! \times \! 2$ subgrids of the grid should contain at least one obstacle. Bob wants to minimize the number of grid cells where he places obstacles.

Help Bob count the number of ways he can place the minimum number obstacles to prevent Alice from placing her block. Output this number modulo a prime number $p$. Note that the answer is not the minimum number of obstacles, but rather the count of the number of ways Bob can place the minimum number of obstacles. For example, if $n=m=2$ for a $2\! \times \! 2$ grid, Bob only has to place $1$ obstacle, but there are $4$ ways to place it, so the answer in this case is $4$.

-----Input-----
The single line of input contains three space-separated integers $n$ ($2 \leq n \leq 25$), $m$ ($2 \leq m \leq 10^3$) and $p$ ($10^8 \leq p \leq 10^9+7$, $p$ is a prime number), where Alice’s grid is of size $n\! \times \! m$, and $p$ is a large prime modulus.

-----Output-----
Output a single integer, which is the number of ways Bob can place the minimum number of obstacles in the $n\! \times \! m$ grid to prevent Alice from placing her $2 \! \times \! 2$ block. Since this may be very large, output it modulo $p$.

-----Examples-----
Sample Input:
4 4 999999937
Sample Output:
79
user: Me and my friends are ordering a big pizza to share. As you can imagine this is quite complicated, since everyone has different wishes about what should be on the pizza. For instance Gunnar wants bananas on the pizza, Emma doesn’t want bananas but wants olives, Marc wants there to be tomatoes, and so on. Fortunately, against all odds, we managed to come up with a selection of toppings such that everyone had at least $2/3$’s of their wishes fulfilled, which we unanimously decided was good enough.

But then, disaster struck! We sent out Lukáš to buy the pizza, but he accidentally lost the piece of paper on which we had written down our carefully selected list of toppings. Now we’re back at square one, and have to construct a new selection of toppings. Given how long it took us to find the original selection of toppings, we have decided to lower our standards a bit and just try to find a selection such that everyone has strictly more than $1/3$ of their wishes fulfilled.

Can you help us with this? If you do, you’ll get some pizza!

-----Input-----
The first line of input contains an integer $1 \le N \le 10000$, the number of friends in the group (including yourself). Each of the next $n$ lines contains the list of wishes of one of the friends. This list starts with an integer $1 \le w \le 30$, the number of wishes this friend has, followed by a space-separated list of wishes. Each wish is either “+<topping>” or “-<topping>” where <topping> is the name of a topping, indicating that this friend wants or does not want this topping. Each topping name appears at most once in each list.

Topping names are non-empty strings of up to $15$ lower-case English letters ‘a’-‘z’. There are at most $250$ different toppings.

-----Output-----
Output a list of toppings (without repetitions, separated by spaces or newlines) such that each friend has strictly more than $1/3$ of their wishes fulfilled. You may assume that there exists a list such that every friend has at least $2/3$ of their wishes fulfilled.

Your list of toppings is not allowed to contain any toppings that are not mentioned in the input, nor is it allowed to contain repetitions.

-----Examples-----
Sample Input:
1
4 +zucchini +mozzarella +mushrooms -artichoke
Sample Output:
zucchini
mozzarella
mushrooms
artichoke
user: Curse your rival! Every year at the annual Rock Paper Scissors tournament, you have made it to the final match. (Your Rock technique is unmatched, and your Paper cuts to the bone! Your Scissors need a little work, though.) But every year, he defeats you, even though his moves appear entirely random! And he claims to the press that he simply cannot be beaten. What is his secret?

Fortunately, you think you have figured it out. This year, just before the tournament, you caught him visiting various shamans around town. Aha! He is using the supernatural against you! You figured two can play at this game. So you went and visited a set of fortune-tellers, who have each used a Tarot deck to predict a sequence that your rival will end up using, sometime during the match.

However, your initial excitement has passed, and now you are feeling a little silly. This cannot possibly work, right? In the end it feels like you have paid good money for a fraudulent, random set of predictions. Oh well; you might as well keep an eye out for some of them during the match. But which predictions will you use?

In the final match, you and your rival will play $n$ rounds of Rock Paper Scissors. In each round, your rival and you will both choose one of the three options (Rock, Paper, or Scissors). Based on your selections, a winner of the round will be determined (exactly how is irrelevant to this problem).

Given the length of the final match and the various predictions, sort them in order of how likely they are to appear sometime during the match as a contiguous sequence of options chosen by your rival, assuming he is choosing his symbol in each round independently and uniformly at random.

-----Input-----
The first line of input contains two integers $n$ ($1\leq n \leq 10^6$), the number of rounds in the final match, and $s$ ($1 \leq s \leq 10$), the number of sequences. The remaining $s$ lines each describe a prediction, consisting of a string of characters ‘R’, ‘P’, and ‘S’. All predictions have the same length, which is between $1$ and $n$ characters long, inclusive, and no longer than $10^5$.

-----Output-----
Display all of the predictions, sorted by decreasing likelihood of appearance sometime during the final match. In the case of tied predictions, display them in the same order as in the input.

-----Examples-----
Sample Input:
3 4
PP
RR
PS
SS
Sample Output:
PS
PP
RR
SS
user: About to leave the university to go home, you notice dark clouds packed in the distance. Since you’re travelling by bicycle, you’re not looking forward to getting wet in the rain. Maybe if you race home quickly you might avert the rain. But then you’d get wet from sweat…

Facing this dilemma, you decide to consider this problem properly with all data available. First you look up the rain radar image that shows you precisely the predicted intensity of rainfall in the upcoming hours. You know at what time you want to be home at the latest. Also, you came up with a good estimate of how much you sweat depending on your cycling speed. Now the question remains: what is the best strategy to get home as dry as possible?

The rain is given for each minute interval in millilitres, indicating how wet you get from cycling through this — note that you can cycle just a fraction of a whole minute interval at the start and end of your trip: then only that fraction of the rain during that interval affects you. Sweating makes you wet at a rate of $s = c \cdot v^2$ per minute where $v$ is your speed in $\mathrm{km}/\mathrm{h}$ and $c$ is a positive constant you have determined. You have to cover the distance to your home in a given time (you don’t want to wait forever for it to become dry), but otherwise you can choose your strategy of when to leave and how fast to cycle (and even change speeds) as you wish. What is the least wet you can get from the combination of rain and sweat?

-----Input-----
 - One line containing a single positive integer $T$ ($0 < T \le 10000$), the number of minutes from now you want to be home by at the latest.
 - Another line with two positive floating point numbers: $c$ ($0.01 \le c \le 10$), the constant determining your sweating, and $d$ ($1 \le d \le 50$), the distance from university to home in kilometres.
 - $T$ more lines, where each line contains an integer $r_ i$ ($0 \le r_ i \le 100$) the number of millilitres of rain during the $i$-th minute interval (zero-based).

-----Output-----
On a single line print a floating point number: the number of millilitres of rain and sweat you get wet from when optimally planning your cycle home. Your answer should be correct up to an absolute or relative precision of $10^{-6}$.

-----Examples-----
Sample Input:
5
0.1 2.0
0
0
0
0
0
Sample Output:
288
user: Strings of yarn have been popular in Catland for ages. Which cat has not spent many a lazy afternoon bouncing around a ball of yarn? Lately however, strings of yarn have gotten competition: strings of characters. It turns out that these are almost as much fun as yarn, and generally much safer as well (so far, no cat has had to call 911 on account of any character string-related entanglement accidents). 

Naturally, some strings are more stylish than others, and for cool cats it is important to engage in their string-playing pastime with style. The meow factor of a string $S$ is the minimum number of operations needed to transform $S$ into a string $S’$ which contains the word “meow” as a substring, where an operation is one of the following four:
 - Insert an arbitrary character anywhere into the string.
 - Delete an arbitrary character anywhere from the string.
 - Replace any character in the string by an arbitrary character.
 - Swap any two adjacent characters in the string.

Write a program to compute the meow factor of a string of characters.

-----Input-----
The input consists of a single line containing a string $S$, consisting only of lower-case letters ‘a’-‘z’. The length of $S$ is at least $1$ and at most $10^6$.

-----Output-----
Output the meow factor of $S$.

-----Examples-----
Sample Input:
pastimeofwhimsy
Sample Output:
1
user: A group of friends was given a necklace. The necklace is a circular nylon wire with gold beads of various weights glued on. They want to cut the wire into segments so that everybody gets one segment, every person gets the same amount of gold, no bead is cut, and no gold is left over. They wonder whether such a split is possible.

-----Input-----
The first line contains two integers $k$ and $n$ where $k$ is the number of friends and $n$ is the number of beads on the necklace. The next line contains $n$ positive integers—the weights of the beads in the order they occur on the necklace. You may assume $k\leq 1000000$, $n\leq 10000000$, and that bead weights are each $\leq 1000000000$.

-----Output-----
The output consists of a single line consisting of the string YES if the necklace can be split into $k$ segments of equal weight, and the string NO otherwise.

-----Examples-----
Sample Input:
3 4
1 2 2 1 
Sample Output:
YES
user: You are given an undirected tree1 with each of its node assigned a magic $X_ i$. The magic of a path2 is defined as the product of the magic of the nodes on that path divided by the number of the nodes on the path. For example, the magic of a path that consists of nodes with magic $3$ and $5$ is $7.5$ ($3\cdot 5 / 2$). In the given tree, find the path with the minimal magic and output the magic of that path.

-----Input-----
The first line of input contains the integer $N$ ($1 \leq N \leq 10^6$), the number of nodes in the tree. Each of the following $N - 1$ lines contains two integers, $A_ i$ and $B_ i$ ($1 \leq A_ i, B_ i \leq N$), the labels of nodes connected with an edge. The $i$-th of the following $N$ lines contains the integer $X_ i$ ($1 \leq X_ i \leq 10^9$), magic of the $i$-th node.

-----Output-----
Output the magic of the path with minimal magic in the form of a completely reduced fraction $P/Q$ ($P$ and $Q$ are relatively prime integers).

In all test cases, it will hold that the required $P$ and $Q$ are smaller than $10^{18}$.

-----Examples-----
Sample Input:
2
1 2
3
4
Sample Output:
3/1
user: Oh no! As a result of recent elections the “Random Laws” party took control of the government. This is going to have bad consequences for Mr. Bourgeois’ company, which just approved the way new salaries will be calculated. The company has $N$ workers and the salary for worker $i$ is going to be determined as a number drawn uniformly from the range $[L_ i, R_ i]$. Since the company already figured out which workers are the most efficient ones, for each $i$ in $[2, N]$, we know that $L_{i-1} \leq L_ i$ and $R_{i-1} \leq R_ i$, but note that as a result of chance, worker $i-1$ might still end up with a larger salary than worker $i$.

The new government introduced a law, where any worker who got a smaller salary than a coworker can sue the company for the amount of their difference. What’s even more atrocious is that they can do it for every worker who got a larger salary. So if there were three employees: Alice, Bob, and Charlie, who got salaries of $1$, $3$, and $7$ coins respectively, then employee Bob can sue with regards to Charlie for $4$ coins, while Alice can sue for $2$ coins because of Bob and for $6$ coins because of Charlie. The total amount of damages the company will have to pay is $12$.

While the exact salary amounts are not known yet, Mr. Bourgeois would like to find out the expected amount of damages that his company will have to pay. Since the answer can be very big, output the answer divided by $N^2$.

-----Input-----
The first line contains $N$, ($1 \leq N \leq 100000$). The next $N$ lines each contain two real numbers $L_ i$ and $R_ i$ ($1 \leq L_ i \leq R_ i \leq 10^6$). All real numbers in the input have at most $6$ digits after the decimal point.

-----Output-----
Output one number: expected payment divided by $N^2$. Your answer will be considered correct if its absolute or relative error is less than $10^{-4}$.

-----Examples-----
Sample Input:
2
1.2 10.2
2.2 15.2
Sample Output:
1.114672365
user: Amazin’ Inc, an up-and-coming company in e-commerce, has recently optimized its operations to make the most out of its workers. Thanks to state-of-the-art prediction methods, Amazin’ now knows in advance how many workers will be needed each day for the foreseeable future. Using this information they can adjust the size of their workforce on a day-to-day basis by firing and/or hiring workers so that they always have exactly as many as are needed each day. In order to prevent the workers from getting too comfortable and organizing themselves, they will also regularly fire workers and replace them with new ones. For instance, if on some day four more workers are needed than yesterday, Amazin’ might fire $10$ people and then hire $14$ new ones on that day. 

Unfortunately, due to labor laws, the firing of workers must follow a last-in-first-out order: the people who have been employed the shortest time must be fired first. Furthermore, a fired person cannot be re-hired within the foreseeable future so it is not possible to circumvent the law by firing some people and then immediately re-hiring some of them.

But this story is actually about HR, not workers! Every day, one employee from the HR department is assigned to be responsible for giving the fired workers the bad news that they are fired, and for then giving the newly hired workers the good news that they are hired. In order to minimize work environment problems in the form of social awkwardness for the HR staff, a policy has been established requiring that the HR person firing an employee must always be a different HR person than the one welcoming them when they were hired.

Now the time has come for the HR department to also optimize itself, by making itself as small as possible. Unlike workers, new HR staff cannot be hired with short notice, so the HR personnel must be permanent employees. What is the smallest number of HR people needed in order to manage all the planned hirings and firings?

-----Input-----
The first line of input contains an integer $n$ ($1 \le n \le 10^5$), the length in days of the foreseeable future. Then follow $n$ lines, the $i$th of which contains two integers $f_ i$ and $h_ i$ ($0 \le f_ i, h_ i \le 10^6$) where $f_ i$ is the number of workers fired on day $i$ and $h_ i$ the number of people hired.

The number of workers fired on a day is never larger than the number of currently employed workers (in other words, $f_ i \le \sum _{j=0}^{i-1} h_ j-f_ j$ for all $1 \le i \le n$).

-----Output-----
Output a line with an integer $k$, the smallest number of HR people needed. The HR people are arbitrarily given IDs from $1$ to $k$. Then output a line with $n$ integers, the $i$th of which contains the ID of the HR person in charge of the firing and hiring on day $i$. If there is more than one solution, any one will be accepted.

-----Examples-----
Sample Input:
4
0 3
1 1
2 1
2 0
Sample Output:
3
1 2 3 2
user: Alf and Beata were two young adults living together a long, long time ago, before you could spend all your afternoons competing in programming. Their lives were thus much more boring than those of today’s young adults. How could you even survive back then, you might ask yourself. The answer is simple: you write down numbers on pieces of paper! Our two cohabitants loved writing integers on papers, and often had huge piles of them each afternoon. To avoid filling their entire living room with integers, Beata challenged her friend to a game every evening to determine who should take the trash out – the Guessing Circle game.

The Guessing Circle game is played by two players (in our case, Alf and Beata) using a large circle of $n$ pieces of papers, each paper labelled with some integer. Alf starts by choosing an integer $x$ that appears on some piece of paper. Beata then tries to figure out what this integer is by asking a series of questions. In each question, Beata picks an integer $y$ that appears on a piece of paper and asks if $y$ is closest to $x$ on the circle when going clockwise or counter-clockwise (measured in the number of pieces of paper between them). If both directions give the same distance, for instance if $y = x$, Alf can choose which one of the two possible answers to provide.

They had initially agreed that no two pieces of paper may have the same integer written on them, but Alf found this hugely unfair – it was quite easy for Beata to figure out $x$. Instead, he suggested a variant where some integers can appear multiple times in the circle. When providing the answer to a question $y$ from Beata, he is instead allowed to choose any pair of papers on which $x$ and $y$ appear, and give his answer for these two papers.

Beata reluctantly agreed to play the new variant, as long as Alf promises to choose an integer in the circle such that Beata can eventually figure it out. Knowing which these integers are turned out to be quite a tricky task for Alf, and he often had to spend hours before the game proving that Beata would be able to deduce which number he had chosen. Write a program to help Alf determine which numbers he can choose.

-----Input-----
The first line of integers contains $n$ ($2 \le n \le 15000$), the number of pieces of paper in the circle. The next line contains $n$ integers, the integers written on the pieces of paper, each between $1$ and $15000$. They are given clockwise in the order they appear on the circle, and are not necessarily unique.

-----Output-----
Output all integers $x$ that Alf can choose in the game such that given enough guesses, Beata can uniquely determine the value of $x$. List these values in increasing order. If there is no such integer $x$, output “none”.

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

Sample Input 2:
3
1 1 2
Sample Output 2:
none

Sample Input 3:
4
1 2 1 3
Sample Output 3:
none