meetforever/GRPO_CL huggingface dataset:
user: Young Luka is an art dealer. He has $N$ clients and sells artistic paintings to each client. Each client can purchase either colored paintings or black and white paintings, but not both. The client denoted with $i$ wants to purchase at most $a_ i$ colored paintings and at most $b_ i$ black and white paintings. The client will always purchase at least one paintings. Luka has an almost unlimited amount of paintings, so the number of paintings required from the clients is never a problem. Luka doesn’t like selling black and white paintings and knows that if less than $C$ people get colored paintings, it will make him feel sad.

His clients constantly keep changing their requests or, in other words, the number of paintings they want to purchase. Because of this, Luka is often troubled by the question: “How many different purchases are there, so that at least $C$ clients get at least one colored painting?” Help Luka and save him from his worries.

-----Input-----
The first line of input contains two integers $N$, $C$ ($1 \leq N \leq 100\ 000$, $1 \leq C \leq 20$). The second line of input contains $N$ integers $a_ i$ ($1 \leq a_ i \leq 1\ 000\ 000\ 000$). The third line of input contains $N$ integers $b_ i$ ($1 \leq b_ i \leq 1\ 000\ 000\ 000$).

The fourth line of input contains the number of requirement changes $Q$ ($1 \leq Q \leq 100\ 000$). Each of the following $Q$ lines contains three integers, the label of the person changing the requirements $P$ ($1 \leq P \leq N$), the maximal number of colored paintings they want to purchase $a_ P$ ($1 \leq a_ P \leq 1\ 000\ 000\ 000$) and the maximal number of black and white paintings they want to purchase $b_ P$ ($1 \leq b_ P \leq 1\ 000\ 000\ 000$).

-----Output-----
The output must consist of $Q$ lines where each line contains the number of different purchases modulo $10\ 007$.

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

Sample Input 2:
2 2
1 2
2 3
2
1 2 2
2 2 2
Sample Output 2:
4
4
user: One day you were visiting a museum where you saw a modern piece of art that shows a grid of $N$ rows and $M$ columns. Each cell in the grid has a color, which can be one of the four: red (‘R’), green (‘G’), blue (‘B’), and white (‘W’).

You think that the art is so simple so that even you can reproduce it yourself. After getting home, you get yourself a grid board with exactly $N$ rows and $M$ columns. The cells of the board are initially all white. You also find some red, green and blue paint (unfortunately you didn’t find any white paint), as well as a square stamp of a size equal to exactly $3 \times 3$ cells. With the tools at hand, in each step you can choose a square of $3\times 3$ cells from the board and pick a color, and then stamp the chosen square of cells with that color (overwriting their previous colors). Note that the chosen square is not allowed to exceed the board boundary, otherwise you would spill the paint and make your home messy.

You soon realize that the art seems to be simple but is not so easy to reproduce. Actually, you are not even sure about whether it is possible to reproduce the art with the aforementioned procedure. You eventually decide to make it a ProgNova question and ask some smart people for help.

-----Input-----
The first line contains two integers $N$ and $M$ ($3 \leq N, M \leq 30$). Each of the next $N$ lines has a string of $M$ characters. All strings contain only letters ‘R’, ‘G’, ‘B’, and ‘W’, representing the colors of the cells in the art piece you want to reproduce.

-----Output-----
Output “YES” if it is possible to reproduce the art piece, or “NO” otherwise.

-----Examples-----
Sample Input 1:
4 5
WRRRG
WRRRG
WRRRG
WBBBB
Sample Output 1:
YES

Sample Input 2:
3 4
WWRR
WRRR
WRRR
Sample Output 2:
NO
user: We always hope things in our lives will run smoothly, and having smooth arrays may help. An array $A$ of $N$ non-negative integers is $K_ S$-smooth if the sum of every set of $K$ consecutive integers is exactly $S$. Unfortunately, not all arrays are $K_ S$-smooth. In fact, all $K_ S$-smooth arrays must contain a repeating pattern of length $K$. The image to the right shows an array of smoothies, and is totally unrelated to this problem, but may help you relax. 

Any array can be made $K_ S$-smooth by changing its elements. In each change one element may be modified to any integer between $0$ and $S$, inclusive. You want to make all of your arrays smooth, but you don’t want to make any more changes than necessary. So the question is: What is the minimum number of changes you have to make so that a given array would become $K_ S$-smooth?

-----Input-----
The first line of input will consist of three integers of the form:\[ N\; \; K\; \; S \]

where $N$ is the size of the array, The remainder of the file will consist of $N$ integers, $a_ n\in A$, separated by white space. All input will be within the following constraints:\[ 1 \le K \le N \le 5000 \]\[ \forall a_ n \in A,\; 0 \le a_ n \le S \le 5000 \]

-----Output-----
Your program must output a single integer specifying the minimum number of changes that must be made in order to make the array $K_ S$ smooth.

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

Sample Input 2:
6 3 5
1
2
3
3
2
1
Sample Output 2:
3
user: Somewhere in an animal kingdom far from here there is a large forest. Inside the forest live a large number of frogs. Every year the frogs gather along the big central road in the forest to show the other animals their unique ability during the Big Animal Prowess Conference (BAPC). 

These frogs are experts in forming towers by climbing on top of each other. They are, however, not experts in gathering orderly on the road, so the frogs have arrived on different positions along the central road. The frogs are also notorious show offs: their every jump is as far as they can and always a prime distance. Not every frog is as strong as the others, so jumping distances may vary. Naturally, the frogs only jump to increase their position, never the other way!

The frog king wants to invite all visitors of the BAPC to marvel at the most spectacular frog tower. Multiple frog towers can be created, but the king wants to show the largest tower at the smallest possible position. He doesn’t want anyone to miss the action because they were at the wrong spot! Can you help the frog king determine the position and size of the tower?

-----Input-----
 - On the first line one integer $n$, the number of frogs gathering on the central road, with $1 \leq n \leq 40$.
 - Then follow $n$ lines with integers $x_ i$ and $d_ i$, the initial position and prime jumping distance of the $i^{th}$ frog. Here $0 \leq x_ i \leq 2^{60}$ and $2 \leq d_ i \leq 10^{8}$. It is given that the product of all unique jumping distances is less than $10^{9}$.

-----Output-----
Output a single line with two integers indicating:
 - the smallest position of the highest frog tower,
 - the size of the highest frog tower.

Separate these integers by a space.

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

Sample Input 2:
5
0 2
1 3
3 3
7 5
9 5
Sample Output 2:
12 3
user: Bar Code is a puzzle game invented by Thinh Van Duc Lai in 2017 that was first published in the New York Times. Bar Code is played on a square grid consisting of $n \times n$ unit squares with $(n+1) \times (n+1)$ grid points. For a given specification, the puzzle requires the marking of some of the borders of these unit squares by drawing a vertical or horizontal bar, subject to the following conditions:
 - For each row/column, the specification describes exactly how many separate groups of consecutive bars of the same orientation (vertical and horizontal, resp.) there should be in this row/column. For instance, if the specification demands 2 1 1, then there have to be $3$ groups of $2$, $1$, and $1$ bars, separated by at least one unmarked border. For example, for a $n = 6$ puzzle with $7$ borders in each row, the following markings would meet this specification:

    1101010 1101001 1100101 0110101

where 1 denotes the presence of a bar and 0 denotes a border that is unmarked.
 - No $2$ bars may touch.

The illustration shows the solution for sample input $3$.

Find a marking that is consistent with the given specification!

-----Input-----
The input consists of a single test case. The first line contains a single integer $n$ ($0 < n \le 9$). This is followed by $n$ lines. The $i^{\text {th}}$ line contains $1$ or more non-negative integers denoting the sizes of groups that must be formed by the vertical bars in row $i$ (counting from the top). If the line contains the single integer $0$, then no borders may be marked for that row. Otherwise, none of the integers on the line will be $0$.

Following that will be another $n$ lines. The $j^{\text {th}}$ line contains $1$ or more non-negative integers denoting the sizes of groups that must be formed by the horizontal bars in column $j$ (counting from the left). If the line contains the single integer $0$, then no borders may be marked for that column. Otherwise, none of the integers on the line will be $0$.

-----Output-----
Output the solution as follows. On the first $n$ lines, output a string of length $n+1$ consisting of 1 and 0 characters. The $i^{\text {th}}$ string should contain a 1 in position $j$ if and only if the $j^{\text {th}}$ vertical border in row $i$ should be marked as a bar. On the next $n+1$ lines, output a string of length $n$ also consisting of 1 and 0 characters. The $i^{\text {th}}$ string should contain a 1 in position $j$ if and only if the $i^{\text {th}}$ horizontal border in column $j$ should be marked as a bar. Rows are counted top down and columns are counted left to right.

If there are multiple solutions, you may output any of them! You may assume that at least one marking exists that is consistent with the specification.

-----Examples-----
Sample Input 1:
2
1
0
0
3
Sample Output 1:
100
000
01
01
01

Sample Input 2:
3
0
1 1
1
1 1
1
1
Sample Output 2:
0000
1001
0010
101
010
000
100
user: Ada, Bertrand and Charles often argue over which TV shows to watch, and to avoid some of their fights they have finally decided to buy a video tape recorder. This fabulous, new device can record $k$ different TV shows simultaneously, and whenever a show recorded in one the machine’s $k$ slots ends, the machine is immediately ready to record another show in the same slot.

The three friends wonder how many TV shows they can record during one day. They provide you with the TV guide for today’s shows, and tell you the number of shows the machine can record simultaneously. How many shows can they record, using their recording machine? Count only shows that are recorded in their entirety.

-----Input-----
The first line of input contains two integers $n,k$ ($1\leq k < n \leq 100\ 000$). Then follow $n$ lines, each containing two integers $x_ i,y_ i$, meaning that show $i$ starts at time $x_ i$ and finishes by time $y_ i$. This means that two shows $i$ and $j$, where $y_ i = x_ j$, can be recorded, without conflict, in the same recording slot. You may assume that $0 \leq x_{i} < y_{i} \leq 1\ 000\ 000\ 000$.

-----Output-----
The output should contain exactly one line with a single integer: the maximum number of full shows from the TV guide that can be recorded with the tape recorder.

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

Sample Input 2:
4 1
1 3
4 6
7 8
2 5
Sample Output 2:
3
user: My colleague Elisabeth is lazy, both at work and when going to work. She never wants to do more than necessary, which also applies to her journey to work. Her goal is to use a minimum amount of energy, which she achieves by braking and accelerating as little as possible. This method applies to all wheel-based transport means she owns.

Elisabeth has earlier optimized her route by trial and error, but now wants your help to find the optimal path. She has provided a map with $N$ junctions and $R$ straight one-way roads between the junctions. If a road is dual way, it is represented as two one-way roads.

Fortunately, Elisabeth often works night shifts, so braking and acceleration are only needed when turning in a junction, as there is no other traffic on the roads. Her goal is to find a route where the maximum turning angle in the junctions is minimized, as this means maximized speed. However, the route can not be too long.

-----Input-----
The first line of input contains three space separated integers $J, R, D$ ($2 \leq J \leq 200, 1 \leq R \leq 40000, 1 \leq D \leq 1000000$). $J$ is the number of junctions, $R$ is the number of one-way roads which are connecting two junctions, and $D$ is the maximum distance, in meters, that Elisabeth wants to travel. The road network is such that there is no path, that Elisabeth may want to use, which has the length $L$ such that $D < L < D (1 + 10^{-6})$.

Then follow $J$ lines with two integers $x$ and $y$, ($-100000 \leq x, y \leq 100000$), the distinct coordinates in meters on a flat ground. Elisabeth lives at junction $1$ and her work is at junction $J$.

After that, $R$ lines follow with two integers $a$ and $b$. This should be interpreted as the one-way road between the one-indexed source and destination junctions $a$ and $b$ ($1\leq a, b\leq J$).

-----Output-----
Output one line with the maximum turning angle of the route that thas the maximum turning angle as low as possible. The turning angle must be output in degrees and have an absolute or relative error at most $10^{-6}$. If there is no such route that is short enough, output “Impossible” on a single line.

-----Examples-----
Sample Input 1:
5 6 500
-100 0
-100 100
0 200
100 100
100 0
1 2
1 3
2 3
3 4
3 5
4 5
Sample Output 1:
90.00000000

Sample Input 2:
5 6 450
-100 0
-100 100
0 200
100 100
100 0
1 2
1 3
2 3
3 4
3 5
4 5
Sample Output 2:
126.86989765
user: Quido has lunch in Hugo’s restaurant every day. He likes the restaurant because all of its prices are expressed as integers, and for each possible price (i.e. $$1$, $$2$, $$3$, etc.) there is at least one beverage and at least one main dish on the menu. Every day there are three entries printed on Quido’s lunch bill: the beverage price, the main dish price, and the total price. Hugo knows of Quido’s interest in computational problems and he offered Quido a free dessert each time his lunch bill meets the following three constraints:
 - the bill is not identical to any of Quido’s previous bills,
 - the price of the beverage is less than the price of the main dish, and
 - the prices listed on the bill cannot mutually use the same digit. In essence, any digit which occurs in any of the entries (beverage, main dish, total) must be different from any of the digits of the other two entries.

Quido is on a budget and he pays the same price for his lunch every day. How many times can he have a free dessert?

-----Input-----
The input consists of a single line with one integer representing the price $P$ which Quido pays for each lunch. The value of $P$ is positive and less than $10^{18}$.

-----Output-----
Output the maximum number of times Quido can have a free dessert at Hugo’s restaurant, provided that the price of his lunch is always $P$. Next, the possible bills which result in a free dessert are listed in ascending order with respect to the beverage price. Each bill consists of the price of the beverage followed by the price of the main dish. For simplicity, the value $P$, which is always the same, is not included in the bill.

If there are more than $5000$ possible bills, then output only the first $5000$ bills (but still report the total number of possible bills before the list of bills).

-----Examples-----
Sample Input 1:
37
Sample Output 1:
4
8 29
9 28
11 26
15 22

Sample Input 2:
30014
Sample Output 2:
7
85 29929
88 29926
785 29229
788 29226
7785 22229
7788 22226
7789 22225
user: Gem Island is a tiny island in the middle of the Pacific Ocean. Until recently, it was known as one of the poorest, but also most peaceful, places on Earth. Today, it is neither poor nor peaceful. What happened?

One sunny morning, not too long ago, all inhabitants of Gem Island woke up to a surprise. That morning, each of them suddenly held one sparkling gem in their hand. The gems had magically appeared overnight. This was cause for much rejoicing – everybody was suddenly rich, they could finally afford all the things they had ever dreamed of, and the name of their island made so much more sense now.

The next morning, one of the inhabitants woke up to another surprise – her gem had magically split into two gems! The same thing happened on each of the following nights, when exactly one of the gems (apparently uniformly at random among all the gems on the island) would split into two.

After a while, the inhabitants of Gem Island possessed a widely varying number of gems. Some had a lot and many had only a few. How come some inhabitants had more gems than others? Did they cheat, were they just lucky, or was something else at work?

The island elders have asked for your help. They want you to determine if the uneven distribution of gems is explained by pure chance. If so, that would greatly reduce tensions on the island.

The island has $n$ inhabitants. You are to determine the gem distribution after $d$ nights of gem splitting. In particular, you are interested in the expected number of gems collectively held by the $r$ people with the largest numbers of gems. More formally, suppose that after $d$ nights the numbers of gems held by the $n$ inhabitants are listed in non-increasing order as $a_1 \ge a_2 \ge \ldots \ge a_ n$. What is the expected value of $a_1 + \dots + a_ r$?

-----Input-----
The input consists of a single line containing the three integers $n$, $d$, and $r$ ($1 \leq n, d \leq 500$, $1 \leq r \leq n$), as described in the problem statement above.

-----Output-----
Display the expected number of gems that the top $r$ inhabitants hold after $d$ nights, with an absolute or relative error of at most $10^{-6}$.

-----Examples-----
Sample Input 1:
2 3 1
Sample Output 1:
3.5

Sample Input 2:
3 3 2
Sample Output 2:
4.9
user: Given intervals $[a, b]$ and $[c, d]$, count the number of ordered pairs of co-prime integers $(x, y)$ such that $a \le x \le b$ and $c \le y \le d$. Coprime integers have no common factor greater than $1$.

-----Input-----
The input consists of a single line of four space-separated integers $a$, $b$, $c$, and $d$. These integers satisfy the bounds ($1 \le a \le b \le 10^7$, $1 \le c \le d \le 10^7$).

-----Output-----
Print a single integer: the number of coprime pairs $(x,y)$ with $a \le x \le b, c\le y \le d$.

-----Examples-----
Sample Input 1:
1 5 1 5
Sample Output 1:
19

Sample Input 2:
12 12 1 12
Sample Output 2:
4
user: A smallest different sequence (SDS) is a sequence of positive integers created as follows: $A_1=r \geq 1$. For $n>1$, $A_ n=A_{n-1}+d$, where $d$ is the smallest positive integer not yet appearing as a value in the sequence or as a difference between two values already in the sequence. For example, if $A_1 =1$, then since $2$ is the smallest number not in our sequence so far, $A_2=A_1+2=3$. Likewise $A_3=7$, since $1, 2$ and $3$ are already accounted for, either as values in the sequence, or as a difference between two values. Continuing, we have $1, 2, 3, 4, 6$, and $7$ accounted for, leaving $5$ as our next smallest difference; thus $A_4=12$. The next few values in this SDS are $20, 30, 44, 59, 75, 96, \ldots $ For a positive integer $m$, you are to determine where in the SDS $m$ first appears, either as a value in the SDS or as a difference between two values in the SDS. In the above SDS, $12, 5, 9$ and $11$ first appear in step $4$.

-----Input-----
Input consists of a single line containing two positive integers $A_1$$m$ ($1 \leq r \leq 100, 1 \leq m \leq 200000000$).

-----Output-----
Display the smallest value $n$ such that the sequence $A_1, \ldots , A_ n$ either contains $m$ as a value in the sequence or as a difference between two values in the sequence. All answers will be $\leq 10000$.

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

Sample Input 2:
1 12
Sample Output 2:
4
user: Your friend Edvin loves languages. A personal favourite of his is speaking the Robber Language. For those of you not familiar with it, it is a language transformation where every consonant is doubled, with an ’o’ put in-between. The rest is kept the same. For example, the word "car" would be transformed into "cocaror". 

Your friend Edvin was going to tell you his password to his favourite website, so that you could check out how awesome it is. Edvin is sure that nobody but him and you speak the Robber Language, so he encrypted it using this simple technique, and wrote it down on a note. After trying the password and realising it does not work, you found out that Edvin was drunk during the transformation. Now you want to find out in how many ways the password might have looked originally.

You are given an encrypted password which Edvin has attempted to translate into the Robber Language. Your task is to determine how many passwords might produce the encrypted password. You may assume that the only thing Edvin did wrong, was when scanning through the word with his eyes, he sometimes missed to transform some consonants using the rule above. Edvin is not deterministic in his failures, i.e. if he missed to transform a consonant once, it is not necessary that he would miss it if it occured again later in the string.

A vowel is considered to be one of the letters "a", "e", "i", "o" and "u". A consonant is any other letter.

-----Input-----
Input consists of one encrypted password, the resulting word from Edvins translation. The word can be as long as $1000000$ characters. It will solely consist of lower case letters a-z.

-----Output-----
Output should consist of one number, the number of ways the password might have looked before Edvin translated it. Since this number may be huge, print the remainder modulo $1000009$.

-----Examples-----
Sample Input 1:
car
Sample Output 1:
1

Sample Input 2:
cocar
Sample Output 2:
2
user: Two thieves (who strongly “recommended” us not to reveal their names) has broken into the central bank of Logland, where the country’s cash reserve is stored. In Logland, the currency has the $k$ denominations $1, 2, 4, 8, \ldots , 2^{k - 1}$, and the bank currently stores $x_ i$ coins of denominations $2^i$.

Thieves live by a very strict honor code, which stipulates that any loot grabbed must be divided evenly between the theives. Unfortunately (for the thieves), this may mean that some loot must be left behind. For example, if the bank contained a single coin of denomination $8$, and two coins of denomination $2$, there is no way they could bring the $8$ coin. Instead, they would have to take one coin of value $2$ each, leaving more than half of the possible loot!

Since neither leaving loot nor solving NP-hard problems are things thieves like to do, they have asked you to compute the minimum value of the loot they must leave behind, in order to be able to split the remaining loot evenly.

-----Input-----
The first line of input contains the integer $1 \le k \le 10^6$ – the number of denominations in Logland. The next line contains the $k$ integers $0 \le x_0, x_1, \dots , x_{k-1} \le 2^{30}$, the number of coins of the denominations $2^0, 2^1, \dots , 2^{k - 1}$.

-----Output-----
Output a single line, the minimum amount of money the thieves must leave behind. Since this number may be very large, output it modulo the prime number $10^9 + 7$.

-----Examples-----
Sample Input 1:
4
0 2 0 1
Sample Output 1:
8

Sample Input 2:
5
1000000 1 1 1 1
Sample Output 2:
0