meetforever/GRPO_CL huggingface dataset:
user: The Birthday Paradox is the name given to the surprising fact that if there are just $23$ people in a group, there is a greater than $50\% $ chance that a pair of them share the same birthday. The underlying assumptions for this are that all birthdays are equally likely (which isn’t quite true), the year has exactly $365$ days (which also isn’t true), and the people in the group are uniformly randomly selected (which is a somewhat strange premise). For this problem, we’ll accept these assumptions.

Consider what we might observe if we randomly select groups of $P=10$ people. Once we have chosen a group, we break them up into subgroups based on shared birthdays. Among many other possibilities, we might observe the following distributions of shared birthdays:
 - all $10$ have different birthdays, or
 - all $10$ have the same birthday, or
 - $3$ people have the same birthday, $2$ other people have the same birthday (on a different day), and the remaining $5$ all have different birthdays.

Of course, these distributions have different probabilities of occurring.

Your job is to calculate this probability for a given distribution of people sharing birthdays. That is, if there are $P$ people in a group, how probable is the given distribution of shared birthdays (among all possible distributions for $P$ people chosen uniformly at random)?

-----Input-----
The first line gives a number $n$ where $1 \le n \le 365$. The second line contain integers $c_1$ through $c_ n$, where $1 \le c_ i \le 100$ for all $c_ i$. The value $c_ i$ represents the number of people who share a certain birthday (and whose birthday is distinct from the birthdays of everyone else in the group).

-----Output-----
Compute the probability $b$ of observing a group of people with the given distribution of shared birthdays. Since $b$ may be quite small, output instead $\log _{10}(b)$. Your submission’s answer is considered correct if it has an absolute or relative error of at most $10^{-6}$ from the judge’s answer.

-----Explanations-----
The first sample case shows $P=2$ people with distinct birthdays. The probability of this occurring is $b = 364/365 \approx 0.9972602740$, and $\log _{10}(b) \approx -0.001191480807419$.

The second sample case represents the third example in the list given earlier with $P=10$ people. In this case, the probability is $b \approx 0.0000489086$, and $\log _{10}(b) \approx -4.310614508857128$.

-----Examples-----
Sample Input:
2
1 1
Sample Output:
-0.001191480807419
user: Matryoshkas are sets of traditional Russian wooden dolls of decreasing size placed one inside the other. A matryoshka doll can be opened to reveal a smaller figure of the same sort inside, which has, in turn, another figure inside, and so on.The Russian Matryoshka Museum recently exhibited a collection of similarly designed matryoshka sets, differing only in the number of nested dolls in each set. Unfortunately, some over-zealous (and obviously unsupervised) children separated these sets, placing all the individual dolls in a row. There are $n$ dolls in the row, each with an integer size. You need to reassemble the matryoshka sets, knowing neither the number of sets nor the number of dolls in each set. You know only that every complete set consists of dolls with consecutive sizes from $1$ to some number $m$, which may vary between the different sets. 

When reassembling the sets, you must follow these rules:
 - You can put a doll or a nested group of dolls only inside a larger doll.
 - You can combine two groups of dolls only if they are adjacent in the row.
 - Once a doll becomes a member of a group, it cannot be transferred to another group or permanently separated from the group. It can be temporarily separated only when combining two groups.

Your time is valuable, and you want to do this reassembly process as quickly as possible. The only time-consuming part of this task is opening and subsequently closing a doll, so you want to minimize how often you do this. For example, the minimum number of openings (and subsequent closings) when combining group [1, 2, 6] with the group [4] is two, since you have to open the dolls with sizes 6 and 4. When combining group [1, 2, 5] with the group [3, 4], you need to perform three openings.

Write a program to calculate the minimum number of openings required to combine all disassembled matryoshka sets.

-----Input-----
The input consists of a single test case. A test case consists of two lines. The first line contains one integer $n$ ($1 \le n \le 500$) representing the number of individual dolls in the row. The second line contains $n$ positive integers specifying the sizes of the dolls in the order they appear in the row. Each size is between $1$ and $500$ inclusive.

-----Output-----
Display the minimum number of openings required when reassembling the matryoshka sets. If reassembling cannot be done (some of the kids might have been excessively zealous and taken some dolls), display the word impossible.

-----Examples-----
Sample Input:
7
1 2 3 2 4 1 3
Sample Output:
7
user: In this problem, you have to analyze a particular sorting algorithm. The algorithm processes a sequence of $n$ distinct integers by swapping two adjacent sequence elements until the sequence is sorted in ascending order. For the input sequence

9 1 0 5 4 ,

Ultra-QuickSort produces the output

0 1 4 5 9 .

Your task is to determine how many swap operations Ultra-QuickSort needs to perform in order to sort a given input sequence.

-----Input-----
Input begins with a line that contains a single integer $1 \le n \le 500000$ – the length of the input sequence. Each of the the following $n$ lines contains a single integer $0 \le a[i] \le 999999999$, the $i$-th input sequence element.

-----Output-----
Prints a single line containing an integer number op, the minimum number of swap operations necessary to sort the given input sequence.

-----Examples-----
Sample Input:
5
9
1
0
5
4
Sample Output:
6
user: Rainfall is measured in millimeters. The rain is collected in a vertical transparent tube with millimeter markings, and once the rain has stopped falling, one can check the height of the water in the tube.

In our problem, the tube unfortunately has a leak at height $L$ millimeters (mm). If the water level is above the leak then water drains from the tube at a rate of $K$ millimeters per hour (mm/h).

We want to figure out how much rain fell during a particular rainfall. We assume that the tube is high enough that it does not overflow. We also assume that rain falls at an (unknown) uniform rate during a rainfall, and that water does not evaporate from the tube. The height of the leak itself is also negligible.

-----Input-----
The input is a line with five positive numbers: $L$ $K$ $T_1$ $T_2$ $H$ where$L$

is where the leak is (mm)$K$

is the rate at which water leaks (mm/h)$T_1$

is the duration of the rainfall (h)$T_2$

is the time between the end of the rainfall and the observation of the water level (h)$H$

is the water level in the tube when we observe it (mm)

Each number is at least $0.01$ and at most $1000.00$, and each is given with exactly two decimals.

-----Output-----
One line with two floating point numbers $F_1$ $F_2$ where $F_1$ is the smallest rainfall in millimeters that would result in the given observation, and $F_2$ is the largest rainfall in millimeters that would result in the given observation. Values with either absolute or relative error smaller than $10^{-6}$ are acceptable.

-----Examples-----
Sample Input:
80.00 0.50 2.00 1.50 80.00
Sample Output:
80.000000 80.759403
user: An ironman triathlon is a race where participants swim for $3.86$ km, ride a bicycle for $180.25$ km, and finally run a marathon, and it is considered one of the toughest sport events. Viveka has been training for an even more challenging competition: the $n$-athlon. In an $n$-athlon race, participants have to go from the starting point to the finishing point through several types of terrain: water, sand, ice, asphalt, etc. To make the race more interesting, participants are free to pick the route that they think suits best their abilities. Last year Viveka achieved an epic victory by skating the last $40$ km in $1$ hour over ice, while her arch-rival Veronica was stuck in a tar pit $1$ m from the finishing point.

The terrain distribution for this year has been published and now it is your task as the optimization expert in Viveka’s team to help her figure out the best route for the race. The competition takes place in a flat area, which we model as the 2D plane, and each type of terrain is shaped as a horizontal strip. Participants are not allowed to leave the race area. You know the position of each strip and Viveka’s speed in that type of terrain.

-----Input-----
The first line contains two pairs of decimal numbers $x_ s$, $y_ s$, $x_ f$, $y_ f$, the $x$ and $y$ coordinates of the starting and finishing point, respectively, in meters. The second line contains one integer $n$ ($1 \leq n \leq 10000$), the number of layers. The third line contains $n-1$ decimal numbers, the $y$ coordinate of each change between layers. Layers are given in order, that is, $y_ s < y_1 < y_2 < \cdots < y_{n-1} < y_ f$, so the shape of layer $i$ is $(-10000,10000)\times (y_{i-1},y_{i})$. The first and last layers extend only until the $y$ coordinate of the starting and finishing point, this is they have shape $(-10000,10000)\times (y_ s,y_1)$ and $(-10000,10000)\times (y_{n-1},y_ f)$ respectively. The fourth line contains $n$ decimal numbers, Viveka’s speed in each layer, in meters per second. All decimal numbers have absolute value at most $10^4$ and at most $4$ decimals.

-----Output-----
Output the minimum time required for Viveka to go from the starting to the finishing point. Your answer should be within absolute or relative error at most $10^{-6}$.

-----Examples-----
Sample Input:
0 0 0 100
2
50
5 1
Sample Output:
60
user: Some viruses replicate by replacing a piece of DNA in a living cell with a piece of DNA that the virus carries with it. This makes the cell start to produce viruses identical to the original one that infected the cell. A group of biologists is interested in knowing how much DNA a certain virus inserts into the host genome. To find this out they have sequenced the full genome of a healthy cell as well as that of an identical cell infected by a virus.

The genome turned out to be pretty big, so now they need your help in the data processing step. Given the DNA sequence before and after the virus infection, determine the length of the smallest single, consecutive piece of DNA that can have been inserted into the first sequence to turn it into the second one. A single, consecutive piece of DNA might also have been removed from the same position in the sequence as DNA was inserted. Small changes in the DNA can have large effects, so the virus might insert only a few bases, or even nothing at all.

-----Input-----
The input consists of two lines containing the DNA sequence before and after virus infection respectively. A DNA sequence is given as a string containing between 1 and $10^5$ upper-case letters from the alphabet {A, G, C, T}.

-----Output-----
Output one integer, the minimum length of DNA inserted by the virus.

-----Examples-----
Sample Input:
AAAAA
AGCGAA
Sample Output:
3
user: The world is made up of $10$ kinds of people, those who understand binary and those who do not. These different kinds of people do not always get along so well. Bob might ask for a $10000$ ounce coffee (meaning binary) and Alice might make misinterpret his request as being in decimal and give him a $10011100010000$ ounce coffee (binary). After Sue explains that this much coffee costs $100$ dollars (decimal), Bob might assume he only has to pay $4$ dollars (interpreting the price as being in binary). In response to these differences that are difficult to resolve, these two groups have divided the world into two regions, the binary-friendly zones and the decimal-friendly zones. They have even published a map like the following to help people keep up with where the areas are (they have used ones and zeros so nobody would have trouble reading it).1111100000

1111000000

1110000011

0111100111

0011111111

Users of binary have to stay in the zones marked with a zero. Users of decimal have to stay in the zones marked with a one. You have to figure out if it is possible for either type of person to get between various locations of interest. People can move north, south, east or west, but cannot move diagonally.

-----Input-----
Input starts with a line containing two positive integers, $1 \le r \le 1000$ and $1 \le c \le 1000$. The next $r$ input lines give the contents of the map, each line containing exactly $c$ characters (which are all chosen from $0$ or $1$).

The next line has an integer $0 \le n \le 1000$. The following $n$ lines each contain one query, given as four integers: $r_1,c_1$ and $r_2,c_2$. These two pairs indicate two locations on the map, and their limits are $1 \le r_1, r_2 \le r$ and $1 \le c_1, c_2 \le c$.

-----Output-----
For each query, output binary if a binary user can start from location $r_1, c_1$ and move to location $r_2, c_2$. Output decimal if a decimal user can move between the two locations. Otherwise, output neither.

-----Examples-----
Sample Input:
1 4
1100
2
1 1 1 4
1 1 1 1
Sample Output:
neither
decimal
user: You administer a large cluster of computers with hard drives that use various file system types to store data. You recently decided to unify the file systems to the same type. That is quite a challenge since all the drives are currently in use, all of them are filled with important data to the limits of their capacities, and you cannot afford to lose any of the data. Moreover, reformatting a drive to use a new file system may significantly change the drive’s capacity. To make the reformat possible, you will have to buy an extra hard drive. Obviously, you want to save money by minimizing the size of such extra storage.

You can reformat the drives in any order. Prior to reformatting a drive, you must move all data from that drive to one or more other drives, splitting the data if necessary. After a drive is reformatted, you can immediately start using it to store data from other drives. It is not necessary to put all the data on the same drives they originally started on – in fact, this might even be impossible if some of the drives have smaller capacity with the new file system. It is also allowed for some data to end up on the extra drive.

As an example, suppose you have four drives $A$, $B$, $C$, and $D$ with drive capacities $6$, $1$, $3$, and $3$ GB. Under the new file system, the capacities become $6$, $7$, $5$, and $5$ GB, respectively. If you buy only $1$ GB of extra space, you can move the data from drive $B$ there and then reformat drive $B$. Now you have $7$ GB free on drive $B$, so you can move the $6$ GB from drive $A$ there and reformat drive $A$. Finally, you move the six total gigabytes from drives $C$ and $D$ to drive $A$, and reformat $C$ and $D$.

-----Input-----
The input begins with a line containing one integer $n$ ($1 \le n \le 10^6$), which is the number of drives in your cluster. Following this are $n$ lines, each describing a drive as two integers $a$ and $b$, where $a$ is the capacity with the old file system and $b$ is the capacity with the new file system.

All capacities are given in gigabytes and satisfy $1 \le a,b \le 10^9$. (One thousand petabytes should be enough for everyone, right?)

-----Output-----
Display the total extra capacity in gigabytes you must buy to reformat the drives.

-----Examples-----
Sample Input:
4
6 6
1 7
3 5
3 5
Sample Output:
1
user: As a new employee at the Northwestern Europe Routing Company (NWERC), you do a lot of thinking about wireless network architectures. Lately you learned about a multi-channel mesh network architecture (called Hyacinth) that equips each mesh network node with multiple network interface cards (NICs) to increase the network throughput. You can choose a channel frequency for each NIC. In order to communicate, for every two network nodes that are in range of each other, their NICs must share at least one common frequency. The theoretical throughput is optimal when the total number of used frequencies in the network is maximal. 

Your bosses at NWERC want you to figure out a procedure for assigning frequencies to the NICs such that the number of frequencies in use is maximized, subject to the constraint that all pairs of adjacent nodes must be able to communicate. A frequency is considered used if any pair of nodes within range of each other share that frequency. In the mesh network that you will be dealing with, each node is equipped with exactly two NICs (i.e., each node can use at most two frequencies). Since you are new at NWERC, your bosses further restrict the network layouts to make your job easier: the network graph will form a tree.

-----Input-----
The input consists of:
 - one line with one integer $n$ ($2 \leq n \leq 10000$), the number of nodes in the network;
 - $n-1$ lines, each with two space-separated integers $i$ and $j$, with $1 \leq i,j \leq n$ signifying that the (one-indexed) network nodes $i$ and $j$ are in range of each other.

-----Output-----
Output a frequency assignment for each of the $2n$ NICs such that all adjacent nodes can communicate and the number of used frequencies is maximized. You should output $n$ lines, where the $i$th line contains the two frequencies of network node $i$. Valid frequencies are nonnegative integers less than $10^9$.

-----Examples-----
Sample Input:
2
1 2
Sample Output:
23 42
42 23
user: You are given a string $s$ consisting only of upper case letters A and B. For an integer $k$, a pair of indices $i$ and $j$ ($1 \le i<j \le n$) is called a $k$-inversion if and only if $s[i] = \textbf{B}$, $s[j] = \textbf{A}$ and $j-i = k$.

Consider the string BABA. It has two $1$-inversions and one $3$-inversion. It has no $2$-inversions.

For each $k$ between $1$ and $n-1$ (inclusive), print the number of $k$-inversions in the string $s$.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The input will consist of a single line with a string $s$, which consists of only upper case As and Bs. The string $s$ will be between $1$ and $1000000$ characters long. There will be no spaces.

-----Output-----
Output $n-1$ lines, each with a single integer. The first line’s integer should be the number of $1$-inversions, the second should be the number of $2$-inversions, and so on.

-----Examples-----
Sample Input:
BABA
Sample Output:
2
0
1
user: Johnny has been roped into a fantasy soccer league and needs your help to set up the best team. 

He has a list of $10$ players that are on his fantasy team (no goalkeepers in this league). He also knows what country they’re from, what league they play in, and what team they play for.

He doesn’t know much about soccer, but he does know these things:
 - If two players are from the same country, a link between the two players will have a synergy score of $1$.
 - If two players are in the same league, a link between the two players will have a synergy score of $1$.
 - If two players are on the same team, a link between the two players will have a synergy score of $2$.
 - If two players are from the same country and in the same league, a link between the two players will have a synergy score of $2$.
 - If two players are from the same country and on the same team, a link between the two players will have a synergy score of $3$.

A team can only be in one league and no two teams will have the same name unless they are the same team.

He has to place the players on his team into a formation of $10$ nodes which can be represented as an undirected graph. The illustration shows the first sample. Therefore, Johnny has to place a player in each node of the graph. Given a particular formation and the members of Johnny’s team, output whether it is possible for Johnny to organize his team to get a perfect team.

A team is a perfect team if and only if every player is placed on a node with a synergy score that is greater or equal to the node’s degree. A node’s degree is the number of links to which it is linked in the formation. A player placed on a node in the formation derives synergy from all players placed on nodes to which the player is linked in the formation. Thus, a node’s synergy score is the sum of the synergy scores of all links to which the node is connected.

-----Input-----
The input will contain a single test case. The first line of the input will have one integer $c$ ($0 \leq c \leq 45$). $c$ represents the number of edges in the formation. The next $c$ lines represent the connections between nodes represented as two integers $a$ ($0 \leq a < 10$) and $b$ ($0 \leq b < 10$), where $a$ is not equal to $b$. Then, the next $10$ lines will be the players on Johnny’s team. Each line will contain four strings in the following order: player name, nation, league, team, which are delimited by a single space. These strings are guaranteed to be non-empty and consist of up to $15$ alphanumeric characters, which includes lower- and uppercase English letters, digits, and hyphens.

-----Output-----
If a perfect team can be organized by Johnny, print yes. Otherwise, print no.

-----Examples-----
Sample Input:
15
0 1
1 2
2 3
0 4
1 5
2 6
3 7
4 5
5 6
6 7
4 8
5 8
6 9
7 9
8 9
Griezmann France LaLiga AtleticoMadrid
Benzema France LaLiga RealMadrid
Ntep France Ligue1 StadeRennais
Sissoko France PremierLeague Spurs
Tolisso France Ligue1 Lyon
Diarra France Ligue1 OM
Evra France CalcioA Juventus
Koscielny France PremierLeague Arsenal
Varane France LaLiga RealMadrid
Sagna France PremierLeague ManCity
Sample Output:
yes