meetforever/GRPO_CL huggingface dataset:
user: Mirko lives in a big enchanted forest where trees are very tall and grow really quickly. That forest can be represented as an $N\cdot N$ matrix where each field contains one tree.

Mirko is very fond of the trees in the enchanted forest. He spent years observing them and for each tree measured how many meters it grew in a year. The trees grow continuously. In other words, if the tree grows $5$ meters in a year, it will grow $2.5$ meters in half a year.

Apart from trees, Mirko likes mushrooms from the enchanted forest. Sometimes, he eats suspicious colorful mushrooms and starts thinking about peculiar questions. Yesterday, this unfortunate thing happened and he wondered what would be the size of the largest connected group of trees that are all of equal height if the trees continue to grow at the same speed they’re growing at that moment.

Mirko quickly measured the current height of all trees in the forest and asked you to answer his question.

Two trees are adjacent if their fields in the matrix share a common edge.

Two trees are connected if there is a sequence of adjacent trees that leads from the first to the second.

A group of trees is connected if every pair of trees in the group is connected.

-----Input-----
The first line of input contains the integer $N$ ($1 \leq N \leq 700$).

After the first line, $N$ lines follow, each of them containing $N$ integers.

The $i$-th line contains integers $h_{ij}$ ($1 \leq h_{ij} \leq 10^6$), the initial height of tree in the $i$-th row and $j$-th column, given in meters.

After that, $N$ more lines follow with $N$ integers each.

The $i$-th line contains integers $v_{ij}$ ($1 \leq v_{ij} \leq 10^6$), the growth speed of the tree in the $i$-th row and $j$-th column, given in meters.

Warning: Please use faster input methods beacuse the amount of input is very large. For example, either set ios::sync_with_stdio(false) or use scanf instead of cin in C++, and use BufferedReader instead of Scanner in Java.

-----Output-----
The first and only line of output must contain the required number from the task.

-----Examples-----
Sample Input:
3
1 2 3
3 2 2
5 2 1
3 2 1
1 2 1
1 2 3
Sample Output:
7
user: Tom is an upcoming senior undergraduate student planning for his courses for the final year. The final year will have two semesters, in which there are $n$ courses offered numbered from $1$ to $n$. Course $i$ has a difficulty level $d_ i$. Two courses may be on a same topic but are at different levels: Level I or Level II. For example, there may be Calculus I and Calculus II offered. In this case, Level I and Level II are two different courses, and Level I is a prerequisite of Level II. Tom can only take Level II in the second semester if he takes Level I in the first semester. He does not necessarily need to take Level II after taking Level I.

To satisfy his graduation requirement, Tom needs to take $k$ courses in total in his last year. Tom would not want to spend too much time studying for courses, as the last year before graduation can be very busy. He therefore would like the sum of difficulties of all courses he takes to be as small as possible. Given that many courses Tom struggles to figure out a good course schedule. Can you help him?

-----Input-----
The first line of the input has two integers $n$ and $k$ ($1 \leq k \leq n \leq 2 \cdot 10^5$). Each of the next $n$ lines has a string and a positive integer. The $i$th line describes the name of course $i$ and its difficulty $d_ i$. All difficulties are no larger than $10^6$. All course names contain at least one and no more than $20$ lowercase English letters. A course name can be optionally followed by a trailing digit ‘1’ or ‘2’ that denotes that it is a Level I or Level II course. It is guaranteed that if the input contains a Level I course, then it also contains its corresponding Level II course, and vice versa. It is also guaranteed that if a course name has two levels, then there does not exist another course with a same name but without levels.

-----Output-----
Output the minimum sum of difficulties of the $k$ courses that Tom can take in the last year.

-----Examples-----
Sample Input:
5 2
linearalgebra 10
calculus1 10
calculus2 20
honorsanalysis1 50
honorsanalysis2 100
Sample Output:
20
user: There are many excellent strategy board games, and your favourite among them is called “Steel Age”. It offers many different paths to victory but you prefer the blood-and-fire-strategy: build as many soldiers as possible and club your opposition into submission. To be able to build soldiers you need two resources: iron ore and coal. 

The board consists of different cells numbered from $1$ to $n$ which can contain resources. The rules for moving from one cell to another are rather complicated: if you can move from cell A to cell B, it does not always mean that you can also move from B to A. For example, if two cells are connected by a river, then you may be able to move downstream, but not upstream, so long as you didn’t invent a steam engine; however, it still could be possible to reach the upstream cell by using roads and taking a detour over other cells.

At the beginning of the game you own only one such cell, where all your settlers are located. At every move you are allowed to move an arbitrary number of settlers from a cell to one of its accessible neighbours. By moving your settlers into a cell for the first time, you “claim” it. Every claimed cell will bind one settler, which has to stay in this cell until the end of the game. However, there is no need to leave a settler in your initial cell because it is where your palace is located and thus the cell stays claimed for all time.

Your goal is to claim at least one cell containing the resource “iron ore” and at least one cell with resource “coal” in order to be able to build soldiers. What is the minimal number of settlers you need to reach this goal?

-----Input-----
The input consists of:
 - One line with three integers $n$ ($2 \le n \le 10^5$), the number of cells on the playing field, $m$ ($1 \le m < n$), the number of cells containing iron ore, and $k$ ($1 \le k < n$), the number of cells containing coal.
 - One line with $m$ distinct integers $o_1, \ldots , o_ m$ ($1 \le o_ i \le n$ for all $1\le i\le m$), where $o_1, \ldots , o_ m$ are the IDs of cells with iron ore.
 - One line with $k$ distinct integers $c_1, \ldots , c_ k$ ($1 \le c_ i \le n$ for all $1\le i\le k$), where $c_1, \ldots , c_ k$ are the IDs of cells with coal.
 - $n$ lines describing the topology of the board. The $j$-th line of this block specifies the accessible neighbours of the $j$-th cell and consists of the following integers:
 - One integer $0\le a \le 10$, the number of cells accessible from cell $j$.
 - $a$ distinct integers $b_1,\dots , b_ a$ ($1 \le b_ i \le n$, $b_ i \ne j$ for all $1\le i\le a$), the IDs of the cells accessible from cell $j$.

It is guaranteed, that no cell contains both resources, iron ore and coal. At the beginning of the game you own only the cell with ID $1$.

-----Output-----
Output the minimum number of settlers needed to claim at least one cell with coal and at least one cell with iron ore. Output “impossible” if it is impossible to own both, coal and iron ore.

-----Examples-----
Sample Input:
3 1 1
2
3
1 2
2 3 1
1 1
Sample Output:
2
user: A farmer has $n$ goats. Coincidentally, he also has $n$ fixed posts in a field where he wants the goats to graze. He wants to tie each goat to a post with a length of rope. He wants to give each goat as much leeway as possible – but, goat ropes are notorious for getting tangled, so he cannot allow any goat to be able to wander into another goat’s territory. What is the maximum amount of rope he could possibly use?

-----Input-----
There will be a single test case in the input. This test case will begin with an integer $n$ ($2 \le n \le 50$), indicating the number of posts in the field. On each of the next $n$ lines will be a pair of integers, $x$ and $y$ ($0 \le x \le 1000$, $0 \le y \le 1000$) which indicate the cartesian coordinates (in meters) of that post in the field. No two posts will be in the same position. You may assume that the field is large enough that the goats will never encounter its border.

-----Output-----
Output a single floating point number, which indicates the maximum amount of rope that the farmer could possibly use, in meters. Output this value with exactly two decimal places, rounded to the nearest value.

-----Examples-----
Sample Input:
2
250 250
250 750
Sample Output:
500.00
user: Yraglac is running a daycare service for dogs. He has $N$ dogs and he would like to take them on a walk. However, before he can walk the dogs, he needs to feed them first.

Yraglac has $M$ feeding bowls. Because the dogs are very well trained, he can instruct a dog to eat from a specific bowl, but he can’t instruct multiple dogs to eat from the same bowl. A dog is considered to be fed after eating from a single bowl. However, different dogs may have different food preferences, so they may eat from the same food bowl at different speeds.

As you know, dogs get impatient if they have to wait for long periods of time. Therefore, Yraglac wants to minimize the total amount of time that dogs spend waiting. More formally, suppose $t$ is the longest amount of time in seconds that any dog spent eating. Yraglac wants to minimize $T=\sum _{i=1}^{N} t-t_ i$, where $t_ i$ is the time in seconds that the $i^\textrm {th}$ dog spends eating.

-----Input-----
The first line contains integers $N$ and $M$, where $2\leq N\leq M\leq 50$.

Each of the next $N$ lines of the input contains $M$ integers each. The $j^\textrm {th}$ integers on the $i^\textrm {th}$ line denotes the $t_{ij}$, the amount of time that the $i^\textrm {th}$ dog will spend on eating food from the $j^\textrm {th}$ bowl. Note that $1\leq t_{ij}\leq 200$.

-----Output-----
Output $T$, the minimum total waiting time in seconds.

-----Examples-----
Sample Input:
2 3
2 100 10
100 1 10
Sample Output:
0
user: Well, it’s time. Andrew has been accumulating file after file on his computer and could never bring himself to delete any single one of them (“You know Algol might make a comeback, so I better not delete any of those files” is one of a large number of his justifications). But he realizes that not only is it wasting a lot of disk space, but it’s making it hard to find anything when he displays his files on the screen as icons.

Because of the sheer number of files that must be gotten rid of, Andrew would like to use as few delete operations as possible. He can delete multiple files at one time if their icons are all clustered together in a rectangular area on his screen by using the mouse to outline a box around them and then hitting delete (an icon is considered in the box if its center is in the box). This also requires that there are no icons in the box of files that he wants to keep. He figures that maybe if he moves file icons around, he can easily put all of the icons into such a rectangular area, perhaps moving some icons out as well.

For example, in the figure below there are three files to delete (black icons) and two to keep (white icons). By moving two of them as shown in the figure on the right, all of the three icons of files to be deleted can be grouped together for one delete operation (note that there are many other ways to move two icons to accomplish this, but no way to do it by just moving one).

 Figure B.1 

Since he must clean out every directory in his file system, he would like to know the following: given a layout of file icons on the screen, what is the minimum number of icons to move so that he can delete all of the appropriate files with one delete command?



-----Input-----
The input will start with four integers $n_ r$ $n_ c$ $n$ $m$ which indicate the number of pixel rows and columns in the screen $(1 \leq n_ r, n_ c \leq 10000)$, the number of file icons on the screen to be deleted $(n)$ and the number of file icons on the screen that should not be deleted $(m)$, where $n+m \leq 100$. After this will be a set of $2(n+m)$ integers indicating the location of the $n+m$ files, the first $n$ of which are the files to be deleted. Each pair of numbers $r$ $c$ will specify the row and col of the upper left corner of the file icon, where $0 \leq r < n_ r$ and $0 \leq c < n_ c$. All icons are 15 pixels high by 9 pixels wide in size and no two icons will be at the same location, though they may overlap, and at least one pixel of the icon must always reside on the screen (both initially and after they’ve been moved). Edges of a delete rectangle lie on pixel boundaries.

-----Output-----
Output the minimum number of file icons that must be moved in order to delete all the appropriate files in one delete operation.

-----Examples-----
Sample Input:
80 50 3 2
75 5 25 20 50 35
50 5 25 35
Sample Output:
2
user: In Gridnavia, three languages are spoken: Arwegian, Banish, and Cwedish. Gridnavia consists of an $n \times m$ grid, where at least one language is spoken in each cell. It is known that each of the three languages is spoken in a non-empty connected subset of grid cells. Connected means that it is possible to get between any pair of cells by moving through adjacent cells, where two cells are said to be adjacent if they share a side. 

You have made a survey to find where in Gridnavia each language is spoken. The following question was sent to every cell in the region: “Please indicate if one or several of the languages is spoken in your cell”. But due to a misprint, there were no choices after the question, so everyone just wrote “one” or “several”. So the only thing you know is for each cell whether exactly one, or more than one language is spoken in that cell.

To make the best out of the situation, you should find any division of the three languages that corresponds to the information.Input

The first line of input contains two integers $n$ and $m$ ($1 \leq n,m \leq 200$).

The following $n$ lines each contain a string of length $m$, consisting of the characters 1 and 2. The $j$th character on the $i$th line is 1 if exactly one language is spoken in that cell, and 2 if at least two languages are spoken.Output

If the languages can be divided according to the information, then output three copies of the grid. The first copy should consist of characters “A” and “.”, where the “A” indicates that Arwegian is spoken in that cell, and “.” indicates that it isn’t spoken. The following two grids should consist of characters “B” and “.”, and “C” and “.”, respectively, with the corresponding information about Banish and Cwedish.

Remember that the three regions have to be connected and non-empty, and every cell must be part of some region. For readability, it is recommended to put empty lines in between the three grids, but it is not necessary to do so. If there are multiple solutions any one will be accepted.

Otherwise, if there is no way to divide the languages, output “impossible”.

-----Examples-----
Sample Input:
3 4
2211
1112
1112
Sample Output:
AAAA
...A
....

BB..
BBBB
...B

....
...C
CCCC
user: Alice: How can it be?

 Rabbit: Trust me Alice. It always takes the same time. When I go from my home up the road to Queen of Hearts’ Castle, my watch counts nine hours. However, if I continue down to Mad Hatter’s House, my watch counts just two hours in total. Isn’t that great?

 Alice: How can it be Rabbit? The path is longer and you take a shorter time to do it? How can it be?

 Rabbit: Trust me Alice! It is all recorded in my logbook. You can check it. All my trips are there...

 Alice: Rabbit, I do not think it can help me...

 Rabbit: Alice, no matter where you are, or where you want to go, or the track you choose, you’ll be able to find how long it takes you.

 Alice: Really?

 Rabbit: For sure!

 


 Poor Rabbit, poor Alice.

White Rabbit is helping Alice finding a quick way home through the Rabbit’s hole with his holy logbook of trips. The problem lies in the chronometer of its bizarre pocket watch (it displays the hours from zero to $12$), and the way the Rabbit counts the time with it: If a journey takes $14$ hours (real time), seeing the pointer resting above number one, he assumes it took one hour.

Given that the White Rabbit is telling the truth, can you help Alice finding how long the shortest path home takes, using the Rabbit’s logbook of trips?

-----Task-----
Your task is to find the shortest real time it takes for Alice to go from her present location to the Rabbit’s hole. For each trip, the White Rabbit wrote down the trip time, the number of visited locations (not necessarily distinct) and the sequence in which they were visited. That sequence defines the trip because there is at most one direct track between any two locations in the Wonderland and it takes the same time both ways. The White rabbit’s logbook contains trips using all roads in Wonderland; there are no direct connections beyond those implied by the trips in the log book.

-----Input-----
The first line contains four integers $N$, $A$, $R$ and $T$, where: $N$ is the number of distinct locations; $A$ identifies the place where Alice is located; $R$ corresponds to the Rabbit’s hole location; and $T$ is the number of trips recorded in White Rabbit’s logbook. All locations are identified by numbers from $1$ to $N$. Each of the next $T$ lines describes a trip logged with format $d p a_{1} a_{2} \cdots a_{p}$, where $d$ is the trip duration (according to White Rabbit), $p$ is the number of locations and $a_{1} a_{2} \cdots a_{p}$ is the sequence of visited locations.

-----Constraints-----

$2$

$\leq $

$N$

$\leq $

$200$

Number of locations

$1$

$\leq $

$T$

$\leq $

$500$

Number of trips in the logbook

$2$

$\leq $

$p$

$\leq $

$800$

Number of (possibly repeated) locations in a trip

$1$

$\leq $

$d_{ij}$

$\leq $

$12$

Real time of the direct track between $a_ i$ and $a_ j$ (if it exists)

There are at most $200$ direct tracks. The input will be constructed in such a way that all (real) trip durations are uniquely determined.

-----Output-----
An integer representing the shortest (real) time it takes for Alice to get home.

-----Examples-----
Sample Input:
3 1 3 3
3 4 1 2 3 2
4 3 1 2 1
1 4 1 2 1 3
Sample Output:
9
user: After conquering Britannia, the great Roman general Agricola decided all of his new cities should take advantage of the natural springs found aplenty. He appointed his advisor Wessus Waterus to try to find a way to get each town a fresh supply of water.

There are many springs and many towns and between each are the natural hills and valleys of Roman Britain. Wessus doesn’t want to waste the Imperial coin. He has been tasked with linking each town to a spring by a series of aqueducts using as little material as possible. Water, as we know, only flows downhill so any aqueduct must go from a higher point to a lower; intervening hills, springs and towns are no problem since they can be tunnelled through and on. The only requirement is that all aqueduct components start and end on hilltops.

Any spring must only serve one town, but the Romans are clever enough to find a way for aqueducts to pass each other. Roman engineering is excellent, but has its limits: aqueducts can only be of a limited length.

-----Input-----
 - One line containing four integers: $n,s,t$ and $q$ where $0 < n \le 500$ is the number of hills, $1 \le s \le 40$ is the number of springs, $1 \le t \le s$ is the number of towns and $q$ ($1 \le q \le 3\cdot 10^6$) is the maximum aqueduct length.
 - $N$ more lines, each giving the space-separated integers $x_ i, y_ i, h_ i$: the coordinates and height of a hill ($0 \le |x|, |y|, h \le 10^6$). These hills are numbered $1$ to $n$ in the order given.
 - One line containing $s$ space-separated integers $i_ j$ ($1 \le i_ j \le n$), each representing the number of a hill on which a spring can be found.
 - One line containing $t$ space-separated integers $i_ j$ ($1 \le i_ j \le n$), each giving the number of a hill on which the town can be found.

Each hill may only have at most one spring or one town.

-----Output-----
Output one line with one real number, denoting the minimum total length of all aqueducts needed to supply each town with fresh water from its own unique spring or IMPOSSIBLE if there is no way to achieve this. Your answer should be correct up to an absolute or relative precision of $10^{-6}$.

-----Examples-----
Sample Input:
6 2 2 8
0 0 6
3 4 7
0 8 8
6 8 8
6 0 6
6 4 8
3 4
1 5
Sample Output:
20.396078
user: Kang the Penguin has forgotten some letters of his password, help him figure them out!

Of course, Kang knows that something as important as a password should be easy to remember, but it also cannot be too short. Thus, he knows that he originally decided to generate his password in the following manner. First, he starts with some non-empty string $S$, consisting of lowercase letters only. Then, he chooses $26$ non-empty strings $T_ a, T_ b, \dots , T_ z$, each consisting of at least two lowercase English letters. He defines a function $f$, which converts each character $i$ to its corresponding string $T_ i$ and concatenates the results. For example, if $T_ a$ was “abc” and $T_ b$ was “cba”, applying $f$ to “aba” would result in the string “abccbaabc”.

Now, he applies $f$ repeatedly to $S$, applying it $K$ times in total. The final result is his password $P = f^K (S)$.

While he remembers most of his password, he has forgotten $M$ of the letters. The $i^\textrm {th}$ letter that he has forgotten is in position $m_ i$ in the string $P$. It is guaranteed that each $m_ i$ will be less than or equal to the final length of the password $|P|$. Help Kang to figure out what the forgotten letters are!

-----Input-----
The $1^\textrm {st}$ line of the input contains a single lowercase string $S$, where $1 \leq |S| \leq 1000000$.

The $2^\textrm {nd}$ line of the input contains $13$ strings $T_ a, T_ b, \dots , T_ m$, separated by spaces, where $2 \leq |T_ a|, |T_ b|, \dots , |T_ m| \leq 50$.

The $3^\textrm {rd}$ line of the input contains $13$ strings $T_ n, T_ o, \dots , T_ z$, separated by spaces, where $2 \leq |T_ n|, |T_ o|, \dots , |T_ z| \leq 50$.

The strings $T_ a, T_ b, \dots , T_ z$ each contains only lowercase English characters (a–z).

The $4^\textrm {th}$ line of the input contains a single integer $K$, where $1 \leq K \leq 10^{15}$.

The $5^\textrm {th}$ line of the input contains a single integer $M$, where $1 \leq M \leq 1000$.

The $6^\textrm {th}$ line of the input contains $M$ integers, the $i^\textrm {th}$ of which is the integer $m_ i$, where $1 \leq m_ i \leq \min (|f^K(S)|,10^{15})$.

-----Output-----
Output $M$ lines, each containing a single lowercase character. The $i^\textrm {th}$ line of the output should contain the letter in the $m_ i^\textrm {th}$ position of the password $P$.

-----Examples-----
Sample Input:
abca
bc cd da dd ee ff gg hh ii jj kk ll mm
nn oo pp qq rr ss tt uu vv ww xx yy zz
1
2
1 8
Sample Output:
b
c