meetforever/GRPO_CL huggingface dataset:
user: Let $s$ be a given string of up to $10^6$ digits. Find the maximal $k$ for which it is possible to partition $s$ into $k$ consecutive contiguous substrings, such that the $k$ parts form a palindrome. More precisely, we say that strings $s_0, s_1, \dots , s_{k-1}$ form a palindrome if $s_ i = s_{k-1-i}$ for all $0\leq i < k$.

In the first sample case, we can split the string 652526 into $4$ parts as 6|52|52|6, and these parts together form a palindrome. It turns out that it is impossible to split this input into more than $4$ parts while still making sure the parts form a palindrome.

-----Input-----
 - A nonempty string of up to $10^6$ digits.

-----Output-----
 - Print the maximal value of $k$ on a single line.

-----Examples-----
Sample Input 1:
652526
Sample Output 1:
4

Sample Input 2:
12121131221
Sample Output 2:
7

Sample Input 3:
123456789
Sample Output 3:
1
user: In the hills of a distant land there is a valley with many villages. The villages are all ruled by a cruel king who demands that each village pay him tribute in gold each year. The villages are required to bring gold to the king as quickly as possible when he demands it.

There is a bandit village in the kingdom. The bandit village has saved no gold for the king, because they spent every bit of gold that they had! What are they going to do? Well, since they’re bandits, when they travel through other villages to get to the castle, they may (or may not) choose to steal the gold from any of those villages on their path to the castle, and present it to the king as their own.

The bandits travel through as few villages as possible to reach the castle. They also have one other consideration-after delivering the gold to the king the bandits must be able to return home. They consider it unsafe to return home traveling through a village from whom they stole gold. The bandits do not otherwise care how long it takes to return home.

Determine the maximum amount of gold the bandits can steal on their way to the king’s castle and still be able to return home safely.

-----Input-----
There will be a single test case in the input. This test case will begin with a line with two integers, $n$ ($3 \le n \le 36$) and $m$ ($n-1 \le m \le n(n-1)/2$), where $n$ is the number of villages, and $m$ is the number of roads. The villages are numbered $1 \ldots n$. Village $1$ is the bandit’s home, and village $2$ holds the king’s castle. On the next line will be $n-2$ space-separated integers $g$ ($1 \le g \le 5000$), which are the amount of gold in each village $3, 4, \ldots , n$, skipping the bandit’s home and the king’s castle. On each of the following $m$ lines will be two integers, $a$ and $b$ ($1 \le a < b \le n$) indicating that there is a road between villages $a$ and $b$. All roads are two-way. All $m$ $(a,b)$ pairs will be unique. There will be a path, direct or indirect, from every village to every other village.

-----Output-----
Output a single integer, indicating the maximum amount of gold that the bandits can purloin and still get safely home.

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

Sample Input 2:
4 4
24 10
1 3
2 3
2 4
1 4
Sample Output 2:
24

Sample Input 3:
6 8
100 500 300 75
1 3
1 4
3 6
4 5
3 5
4 6
2 5
2 6
Sample Output 3:
800
user: Zapray really loves the Algorithmic Problem Solving course, but he is struggling with it. Recently the class studied Longest Increasing Subsequence (LIS). Zapray knows well how to compute the length of LIS for a given sequence. But on the exam paper there appears to be a different question: “We say an element in a sequence is critical, if removing it will cause the sequence’s LIS length to decrease. Can you find all the critical elements in the sequence below?” Zapray soon realizes that this is a much bigger challenge.

-----Input-----
The input starts with a single integer $n$ ($2 \leq n \leq 10^5$), the length of the given sequence. The next line has $n$ integers giving the numbers in the sequence. The given sequence is a permutation in which each integer from $1$ to $n$ appears exactly once.

-----Output-----
Output all critical elements in the sequence in ascending order. If none of the elements are critical, output “-1”.

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

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

Sample Input 3:
4
2 1 4 3
Sample Output 3:
-1
user: Bureaucrats love bureaucracy. This assertion seems fairly obvious but a less obvious observation is the amount of paperwork this means!

When paperwork is complete, a bureaucrat stamps the official document with their official stamp of office. Some bureaucrats are extra thorough and stamp the document multiple times. We are interested primarily in bureaucrats that stamp their documents exactly twice.

A bureaucrat stamp takes up some rectangular area. For example, the below is a bureaucrat stamp:

       ..#..#.. .######. ..#..#..

When the bureaucrat stamps the paper twice it is potentially moved to a different location on the paper, but it is not rotated. The stamp will always be axis aligned. The ’#’ symbol on a stamp covers the paper with black ink at the cell on the paper that is pressed. A ’.’ doesn’t leave any marks on the paper nor does it remove a mark. If a cell of the paper is marked twice it is not discernable from a cell that is marked once.

You will be given a mark on a paper that was stamped twice by a stamp. Your task is to determine the minimum number of nubs (’#’ symbols) that could have possibly been on the original stamp. The paper is guaranteed to be stamped twice by the entire stamp. (All of the stamp will be on the paper in both stampings)

Consider the following mark on paper:

       ..#..#.. .######. .######. ..#..#..

It could have been made with the first stamp example, with the second stamping exactly one cell below the first. Or, it could have been made with a stamp that looks like this, stamped side-by-side:

       ..#.. .###. .###. ..#..

In this case, 8 would be the correct answer.

-----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, $n$ and $m$ ($1 \le n, m \le 500$) indicating the height and width of the mark on the paper. Each of the next $n$ lines will have exactly $m$ characters, consisting only of ‘#’ and ‘.’, representing the mark. Every test case will have at least one ‘#’.

-----Output-----
Output a single line with a single integer, indicating the minimum number of nubs of a bureaucrat’s stamp that could have possibly made the input mark by stamping exactly twice.

-----Examples-----
Sample Input 1:
4 8
..#..#..
.######.
.######.
..#..#..
Sample Output 1:
8

Sample Input 2:
3 3
...
.#.
...
Sample Output 2:
1

Sample Input 3:
2 6
.#####
#####.
Sample Output 3:
5
user: Primonimo is a game played on an $n \times m$ board filled with numbers taken from the range $1 \ldots p$ for some prime number $p$. At each move, a player selects a square and adds $1$ to the numbers in all squares in the same row and column as the selected square. If a square already shows the number $p$, it wraps around to $1$.

The game is won if all squares show $p$. Given an initial board, find a sequence of moves that wins the game!

-----Input-----
The input consists of a single test case. The first line contains three numbers $n \ m \ p$ denoting the number of rows $n$ ($1 \le n \le 20$), the number of columns $m$ ($1 \le m \le 20$), and a prime number $p$ ($2 \le p \le 97$). Each of the next $n$ lines consists of $m$ numbers in the range $1 \ldots p$.

-----Output-----
If a winning sequence of at most $p \cdot m \cdot n$ moves exists, output an integer $k \le p \cdot m \cdot n$ denoting the number of moves in the sequence. Then output $k$ moves as a sequence of integers that numbers the board in row-major order, starting with $1$. If there are multiple such sequences, you may output any one of them. If no winning sequence exists, output -1.

-----Examples-----
Sample Input 1:
4 5 5
2 1 1 1 2
5 3 4 4 3
4 3 3 3 2
3 1 3 3 1
Sample Output 1:
6
19 12 2 18 5 5

Sample Input 2:
3 3 3
3 1 1
1 3 2
3 2 3
Sample Output 2:
13
4 2 6 1 9 7 5 5 7 1 2 3 3

Sample Input 3:
3 2 2
1 2
2 1
1 2
Sample Output 3:
-1
user: You are given a rooted tree of $N$ vertices where each vertex $v_1$ through $v_ N$ is labeled with a non-negative integer $u_1$ through $u_ N$. The vertex $v_1$ is the root.

Define a “jumping path” within the tree to be a sequence $a_1, a_2, \ldots , a_ k$ of length $k$ where $v_{a_ i}$ is an ancestor of $v_{a_ j}$ for all $1 \le i < j \le k$.

Compute two quantities for the given tree:
 - The length $L$ of the longest jumping path where the labels of the vertices are nondecreasing. That is, $u_{a_ i} \le u_{a_ j}$ for all $1 \le i < j \le L$.
 - The number $M$ of jumping paths of length $L$ where the labels of the vertices are nondecreasing. Since this number may be large, give the remainder of $M$ when divided by the prime $11092019$.

-----Input-----
The first line of input contains an integer $N$ denoting the number of vertices in the tree ($1 \leq N \leq 10^6$).

This is followed by $N$ lines of input indicating the labels $u_1$ through $u_ N$. Each label is an integer in the range $[0, 10^6]$.

The remaining $N-1$ lines describe the tree structure. Skipping the root (which has no parent) and starting with $i=2$, line $i$ gives the parent $p_ i < i$ of vertex $v_ i$.

-----Output-----
Print a single line of output with two integers separated by a space. The first integer is $L$, and the second integer is $M$ modulo the prime $11092019$.

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

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

Sample Input 3:
4
1
5
3
6
1
2
3
Sample Output 3:
3 2
user: You are given a rooted tree with $n$ nodes. The nodes are labeled $1$ to $n$, and node $1$ is the root. Each node has a value $v_ i$.

You would like to turn this tree into a heap. That is, you would like to choose the largest possible subset of nodes that satisfy this Heap Property: For every node pair $i$,$j$ in the subset, if node $i$ is an ancestor of node $j$ in the tree, then $v_ i > v_ j$. Note that equality is not allowed.

Figure out the maximum number of nodes you can choose to form such a subset. The subset does not have to form a subtree.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input will contain a single integer $n$ ($1 \le n \le 2 \cdot 10^5$), which is the number of nodes in the tree. The nodes are numbered $1$ to $n$.

Each of the next $n$ lines will describe the nodes, in order from $i=1$ to $n$. They will each contain two integers $v_ i$ and $p_ i$, where $v_ i$ ($0 \le v_ i \le 10^9$) is the value in the node, and $p_ i$ ($0 \le p_ i < i$) is the index of its parent. Every node’s index will be strictly greater than its parent node’s index. Only node 1, the root, will have $p_1=0$, since it has no parent. For all other nodes ($i=2, 3, \ldots , n$), $1 \le p_ i < i$.

-----Output-----
Output a single integer representing the number of nodes in the largest subset satisfying the Heap Property.

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

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

Sample Input 3:
6
3 0
1 1
2 1
3 1
4 1
5 1
Sample Output 3:
5
user: The Transit Authority of Greater Podunk is planning its holiday decorations. They want to create an illuminated display of their light rail map in which each stretch of track between stations can be illuminated in one of several colors. 

At periodic intervals, the controlling software will choose two stations at random and illuminate all of the segments connecting those two stations. By design, for any two stations on the Greater Podunk Railway, there is a unique path connecting the two.

For maximum color and cheer, the display designers want to avoid having two adjacent segments of track lighting up in the same color. They fear, however, that they may have lost track of this guideline in the process of building the display. One of them has gone so far as to propose a means of measuring just how far from that ideal they may have fallen.

-----Description-----
You are given a tree with $n$ nodes (stations), conveniently numbered from $1$ to $n$. Each edge in this tree has one of $n$ colors. A path in this tree is called a rainbow if all adjacent edges in the path have different colors. Also, a node is called good if every simple path with that node as one of its endpoints is a rainbow path. (A simple path is a path that does not repeat any vertex or edge.)

Find all the good nodes in the given tree.

-----Input-----
The first line of input contains a single integer $n$ ($1 \le n \le 50000$).

Each of the next $n-1$ lines contains three space-separated integers $a_ i$, $b_ i$, and $c_ i$ ($1 \le a_ i, b_ i, c_ i \le n$; $a_ i \ne b_ i$), describing an edge of color $c_ i$ that connects nodes $a_ i$ and $b_ i$.

It is guaranteed that the given edges form a tree.

-----Output-----
On the first line of the output, print $k$, the number of good nodes.

In the next $k$ lines, print the indices of all good nodes in numerical order, one per line.

-----Examples-----
(For the first sample, node $3$ is good because all paths that have node $3$ as an endpoint are rainbow. In particular, even though the path $3 \rightarrow 4 \rightarrow 5 \rightarrow 6$ has two edges of the same color (i.e. $3 \rightarrow 4$, $5 \rightarrow 6$), it is still rainbow because these edges are not adjacent.)

-----Examples-----
Sample Input 1:
8
1 3 1
2 3 1
3 4 3
4 5 4
5 6 3
6 7 2
6 8 2
Sample Output 1:
4
3
4
5
6

Sample Input 2:
8
1 2 2
1 3 1
2 4 3
2 7 1
3 5 2
5 6 2
7 8 1
Sample Output 2:
0

Sample Input 3:
9
1 2 2
1 3 1
1 4 5
1 5 5
2 6 3
3 7 3
4 8 1
5 9 2
Sample Output 3:
5
1
2
3
6
7
user: For every positive integer we may obtain a non-negative integer by multiplying its digits. This defines a function $f$, e.g. $f(38) = 24$.

This function gets more interesting if we allow for other bases. In base $3$, the number $80$ is written as $2222$, so: $f_3(80) = 16$.

We want you to solve the reverse problem: given a base $B$ and a number $N$, what is the smallest positive integer $X$ such that $f_ B(X) = N$?

-----Input-----
The input consists of a single line containing two integers $B$ and $N$, satisfying $2 < B \leq 10000$ and $0 < N < 2^{63}$.

-----Output-----
Output the smallest positive integer solution $X$ of the equation $f_ B(X) = N$. If no such $X$ exists, output the word “impossible”. The input is carefully chosen such that $X < 2^{63}$ holds (if $X$ exists).

-----Examples-----
Sample Input 1:
10 24
Sample Output 1:
38

Sample Input 2:
10 11
Sample Output 2:
impossible

Sample Input 3:
9 216
Sample Output 3:
546
user: The Bluewater Geocloud Organization (BGO) has recently developed brand new software able to predict with pinpoint precision at which second any particular cloud will start and stop raining, and with what intensity. There is, however, some uncertainty about how a cloud will move around; for each zip code, each cloud will be over that zip code with some probability.

You have scraped some information about your zip code from the BGO website, and want to use it to plan your walk to the bus stop. You wish to minimize the expected amount of rain that would fall on you. To reach the bus you must get to the bus stop within $t$ seconds from now. You have timed your walking speed to be exactly $1 \frac{m}{s}$.

To complicate matters, some parts of the walk to the bus are covered by roofs where it might be beneficial to make shorts breaks whilst waiting for the worst rain to pass. Your front door (at $d$ meters from the bus stop) is always under a roof – but the bus stop need not be.

-----Input-----
The first line of input is four space-separated integers: $d$ ($1 \leq d \leq 1000$), the distance to the bus stop in meters, $t$ ($d \leq t \leq 10000$) the time until the bus leaves, $c$ ($0 \leq c \leq 1000000$), the number of clouds tracked by BGO, and finally $r$ ($0 \leq r \leq d$), the number of roofs. The next $c$ lines describe the clouds; the $i$’th such line contains four numbers $s_ i$, $e_ i$, $p_ i$ and $a_ i$ describing the $i$’th cloud:
 - $s_ i$ ($0 \leq s_ i < t$) is an integer giving the number of seconds until the cloud starts its raining period,
 - $e_ i$ ($s_ i < e_ i \leq t$) is an integer giving the number of seconds until the cloud ends its raining period,
 - $p_ i$ ($0 \leq p_ i \leq 1$) is a real number (with at most $6$ digits after the decimal point) giving the probability that the cloud is in your zip code during its raining period, and
 - $a_ i$ ($0 \leq a_ i \leq 100$) is an integer indicating the amount of rain the cloud will release during its raining period, given as nm per second.

Finally $r$ roof segments follow, each on its own line; the $j$’th such line contains two integers $x_ j$ and $y_ j$ ($0 \leq x_ j < y_ j \leq d+1$), indicating that there is a roof segment starting at distance $x_ j$ away from home, ending at distance $y_ j$ away from home along the route to the bus stop. Both your home, the bus stop an the entire route between them are in the same zip code. No two roofs overlap, however one roof may start at the same exact location as another ends.

-----Output-----
The output consists of single a real value, the minimum amount of rain in nm you can expect on your route if you reach the bus stop in time. Answers with absolute or relative precision $10^{-5}$ of the actual value will be accepted.

-----Examples-----
Sample Input 1:
20 60 2 1
5 15 0.33333 30
22 60 0.66666 70
0 10
Sample Output 1:
466.662

Sample Input 2:
3 4 2 1
1 3 0.25 8
2 4 0.66667 15
1 2
Sample Output 2:
10.00005

Sample Input 3:
3 4 1 0
0 2 0.25 8
Sample Output 3:
2.0
user: The lunch break just started! Your mother gave you one litre of pink lemonade. You do not like pink lemonade and want blue lemonade instead. Fortunately for you the other children in class are willing to trade with you.

Each of them is willing to offer you any quantity of the virtually infinite amount of lemonade they got from their mother, in exchange for their favourite lemonade, according to some exchange rate. The other children are sitting in a long row in the class room and you will walk along the row, passing each child only once. You are not allowed to walk back! Of course, you want to maximise the amount of blue lemonade you end up with. In case you can obtain more than $10$ litres of blue lemonade, this is more than you will need, and you will throw away any excess (and keep the $10$ litres).

Fortunately, you know in advance what everybody is offering for trade. Your task is to write a program to find the maximum amount of blue lemonade that you can obtain.

-----Input-----
The input consists of the following:
 - One line containing a single integer $0\leq N \leq 10^5$, the number of children in the class room, excluding yourself;
 - $N$ lines, each containing two strings $O, W$ and a floating point number $0.5 < R < 2$, the name of the lemonade that is offered, the name of the lemonade that is wanted, and the exchange rate: for every litre of lemonade $W$ that you trade you get $R$ litres of lemonade $O$ in return. Exchange rates are given with at most $7$ digits after the decimal point.

All strings are guaranteed to have at most $10$ alphanumeric characters.

-----Output-----
Output a single line containing a single floating point number $M$, the maximum amount (in litres) of blue lemonade that you can obtain. In case you could obtain more than $10$ litres, $M$ is considered to be $10$. You are required to output $M$ with absolute precision $10^{-6}$.

-----Examples-----
Sample Input 1:
3
blue pink 1.0
red pink 1.5
blue red 1.0
Sample Output 1:
1.500000000000000

Sample Input 2:
2
blue red 1.0
red pink 1.5
Sample Output 2:
0.000000000000000

Sample Input 3:
4
orange pink 1.9
yellow orange 1.9
green yellow 1.9
blue green 1.9
Sample Output 3:
10.000000000000000
user: The world-renowned Swedish Chef is planning a gourmet three-course dinner for some muppets: a starter course, a main course, and a dessert. His famous Swedish cook-book offers a wide variety of choices for each of these three courses, though some of them do not go well together (for instance, you of course cannot serve chocolate moose and sooted shreemp at the same dinner). 

Each potential dish has a list of ingredients. Each ingredient is in turn available from a few different brands. Each brand is of course unique in its own special way, so using a particular brand of an ingredient will always result in a completely different dinner experience than using another brand of the same ingredient.

Some common ingredients such as pølårber may appear in two of the three chosen dishes, or in all three of them. When an ingredient is used in more than one of the three selected dishes, Swedish Chef will use the same brand of the ingredient in all of them.

While waiting for the meecaroo, Swedish Chef starts wondering: how many different dinner experiences are there that he could make, by different choices of dishes and brands for the ingredients?

-----Input-----
The input consists of:
 - one line containing five integers $r$, $s$, $m$, $d$, $n$, where $1 \le r \le 1000$ is the number of different ingredients that exist, $1 \le s, m, d \le 25$ are the number of available starter dishes, main dishes, and desserts, respectively, and $0 \le n \le 2000$ is the number of pairs of dishes that do not go well together.
 - one line containing $r$ integers $b_1, \ldots , b_ r$, where $1 \le b_ i \le 100$ is the number of different brands of ingredient $i$.
 - $s+m+d$ lines describing the $s$ starter dishes, then the $m$ main dishes, then the $d$ desserts. Each such line starts with an integer $1 \le k \le 20$ denoting the number of ingredients of the dish, and is followed by $k$ distinct integers $i_1, \ldots , i_ k$, where for each $1 \le j \le k$, $1 \le i_ j \le r$ is an ingredient.
 - $n$ lines each containing two incompatible dishes. Each dish is identified by an integer $1 \le j \le s+m+d$, referring to the $j$’th dish given in the input (so $1 \le j \le s$ refers to the starter dishes, $s < j \le s+m$ refers to the main dishes, and $s+m < j \le s+m+d$ refers to the desserts).

Each pair of incompatible dishes in the input consists of two dishes of different types, and any one pair of dishes is listed at most once.

-----Output-----
If the number of different dinner experiences Swedish Chef can make is at most $10^{18}$, then output that number. Otherwise, output “too many”.

-----Examples-----
Sample Input 1:
6 1 1 1 0
2 3 1 5 3 2
2 1 2
3 3 4 5
1 6
Sample Output 1:
180

Sample Input 2:
3 2 2 1 1
2 3 2
1 1
1 2
1 2
1 3
1 1
2 3
Sample Output 2:
22

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