meetforever/GRPO_CL huggingface dataset:
user: You are given an array $a$, consisting of $n$ positive integers.

Let's call a concatenation of numbers $x$ and $y$ the number that is obtained by writing down numbers $x$ and $y$ one right after another without changing the order. For example, a concatenation of numbers $12$ and $3456$ is a number $123456$.

Count the number of ordered pairs of positions $(i, j)$ ($i \neq j$) in array $a$ such that the concatenation of $a_i$ and $a_j$ is divisible by $k$.


-----Input-----

The first line contains two integers $n$ and $k$ ($1 \le n \le 2 \cdot 10^5$, $2 \le k \le 10^9$).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$).


-----Output-----

Print a single integer — the number of ordered pairs of positions $(i, j)$ ($i \neq j$) in array $a$ such that the concatenation of $a_i$ and $a_j$ is divisible by $k$.


-----Examples-----
Input
6 11
45 1 10 12 11 7

Output
7

Input
4 2
2 78 4 10

Output
12

Input
5 2
3 7 19 3 3

Output
0



-----Note-----

In the first example pairs $(1, 2)$, $(1, 3)$, $(2, 3)$, $(3, 1)$, $(3, 4)$, $(4, 2)$, $(4, 3)$ suffice. They produce numbers $451$, $4510$, $110$, $1045$, $1012$, $121$, $1210$, respectively, each of them is divisible by $11$.

In the second example all $n(n - 1)$ pairs suffice.

In the third example no pair is sufficient.
user: There is an integer sequence A of length N whose values are unknown.
Given is an integer sequence B of length N-1 which is known to satisfy the following:
 B_i \geq \max(A_i, A_{i+1}) 
Find the maximum possible sum of the elements of A.

-----Constraints-----
 - All values in input are integers.
 - 2 \leq N \leq 100
 - 0 \leq B_i \leq 10^5

-----Input-----
Input is given from Standard Input in the following format:
N
B_1 B_2 ... B_{N-1}

-----Output-----
Print the maximum possible sum of the elements of A.

-----Sample Input-----
3
2 5

-----Sample Output-----
9

A can be, for example, ( 2 , 1 , 5 ), ( -1 , -2 , -3 ), or ( 2 , 2 , 5 ). Among those candidates, A = ( 2 , 2 , 5 ) has the maximum possible sum.
user: Given are positive integers N, M, Q, and Q quadruples of integers ( a_i , b_i , c_i , d_i ).
Consider a sequence A satisfying the following conditions:
 - A is a sequence of N positive integers.
 - 1 \leq A_1 \leq A_2 \le \cdots \leq A_N \leq M.
Let us define a score of this sequence as follows:
 - The score is the sum of d_i over all indices i such that A_{b_i} - A_{a_i} = c_i. (If there is no such i, the score is 0.)
Find the maximum possible score of A.

-----Constraints-----
 - All values in input are integers.
 - 2 ≤ N ≤ 10
 - 1 \leq M \leq 10
 - 1 \leq Q \leq 50
 - 1 \leq a_i < b_i \leq N ( i = 1, 2, ..., Q )
 - 0 \leq c_i \leq M - 1 ( i = 1, 2, ..., Q )
 - (a_i, b_i, c_i) \neq (a_j, b_j, c_j) (where i \neq j)
 - 1 \leq d_i \leq 10^5 ( i = 1, 2, ..., Q )

-----Input-----
Input is given from Standard Input in the following format:
N M Q
a_1 b_1 c_1 d_1
:
a_Q b_Q c_Q d_Q

-----Output-----
Print the maximum possible score of A.

-----Sample Input-----
3 4 3
1 3 3 100
1 2 2 10
2 3 2 10

-----Sample Output-----
110

When A = \{1, 3, 4\}, its score is 110. Under these conditions, no sequence has a score greater than 110, so the answer is 110.
user: You are given an integer sequence A of length N.
Find the maximum absolute difference of two elements (with different indices) in A.

-----Constraints-----
 - 2 \leq N \leq 100
 - 1 \leq A_i \leq 10^9
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
N
A_1 A_2 ... A_N

-----Output-----
Print the maximum absolute difference of two elements (with different indices) in A.

-----Sample Input-----
4
1 4 6 3

-----Sample Output-----
5

The maximum absolute difference of two elements is A_3-A_1=6-1=5.
user: There are N towns in a coordinate plane. Town i is located at coordinates (x_i, y_i). The distance between Town i and Town j is \sqrt{\left(x_i-x_j\right)^2+\left(y_i-y_j\right)^2}.
There are N! possible paths to visit all of these towns once. Let the length of a path be the distance covered when we start at the first town in the path, visit the second, third, \dots, towns, and arrive at the last town (assume that we travel in a straight line from a town to another). Compute the average length of these N! paths.

-----Constraints-----
 - 2 \leq N \leq 8
 - -1000 \leq x_i \leq 1000
 - -1000 \leq y_i \leq 1000
 - \left(x_i, y_i\right) \neq \left(x_j, y_j\right) (if i \neq j)
 - (Added 21:12 JST) All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
N
x_1 y_1
:
x_N y_N

-----Output-----
Print the average length of the paths.
Your output will be judges as correct when the absolute difference from the judge's output is at most 10^{-6}.

-----Sample Input-----
3
0 0
1 0
0 1

-----Sample Output-----
2.2761423749

There are six paths to visit the towns: 1 → 2 → 3, 1 → 3 → 2, 2 → 1 → 3, 2 → 3 → 1, 3 → 1 → 2, and 3 → 2 → 1.
The length of the path 1 → 2 → 3 is \sqrt{\left(0-1\right)^2+\left(0-0\right)^2} + \sqrt{\left(1-0\right)^2+\left(0-1\right)^2} = 1+\sqrt{2}.
By calculating the lengths of the other paths in this way, we see that the average length of all routes is:
\frac{\left(1+\sqrt{2}\right)+\left(1+\sqrt{2}\right)+\left(2\right)+\left(1+\sqrt{2}\right)+\left(2\right)+\left(1+\sqrt{2}\right)}{6} = 2.276142...
user: The window of Takahashi's room has a width of A. There are two curtains hung over the window, each of which has a horizontal length of B. (Vertically, the curtains are long enough to cover the whole window.)
We will close the window so as to minimize the total horizontal length of the uncovered part of the window.
Find the total horizontal length of the uncovered parts of the window then.

-----Constraints-----
 - 1 \leq A \leq 100
 - 1 \leq B \leq 100
 - A and B are integers.

-----Input-----
Input is given from Standard Input in the following format:
A B

-----Output-----
Print the total horizontal length of the uncovered parts of the window.

-----Sample Input-----
12 4

-----Sample Output-----
4

We have a window with a horizontal length of 12, and two curtains, each of length 4, that cover both ends of the window, for example. The uncovered part has a horizontal length of 4.
user: You are given an integer N.

For two positive integers A and B, we will define F(A,B) as the larger of the following: the number of digits in the decimal notation of A, and the number of digits in the decimal notation of B.

For example, F(3,11) = 2 since 3 has one digit and 11 has two digits.

Find the minimum value of F(A,B) as (A,B) ranges over all pairs of positive integers such that N = A \times B.

-----Constraints-----
 - 1 \leq N \leq 10^{10}
 - N is an integer.

-----Input-----
The input is given from Standard Input in the following format:
N

-----Output-----
Print the minimum value of F(A,B) as (A,B) ranges over all pairs of positive integers such that N = A \times B.

-----Sample Input-----
10000

-----Sample Output-----
3

F(A,B) has a minimum value of 3 at (A,B)=(100,100).
user: Katsusando loves omelette rice.
Besides, he loves crème brûlée, tenderloin steak and so on, and believes that these foods are all loved by everyone.
To prove that hypothesis, he conducted a survey on M kinds of foods and asked N people whether they like these foods or not.
The i-th person answered that he/she only likes the A_{i1}-th, A_{i2}-th, ..., A_{iK_i}-th food.
Find the number of the foods liked by all the N people.

-----Constraints-----
 - All values in input are integers.
 - 1 \leq N, M \leq 30
 - 1 \leq K_i \leq M
 - 1 \leq A_{ij} \leq M
 - For each i (1 \leq i \leq N), A_{i1}, A_{i2}, ..., A_{iK_i} are distinct.

-----Constraints-----
Input is given from Standard Input in the following format:
N M
K_1 A_{11} A_{12} ... A_{1K_1}
K_2 A_{21} A_{22} ... A_{2K_2}
:
K_N A_{N1} A_{N2} ... A_{NK_N}

-----Output-----
Print the number of the foods liked by all the N people.

-----Sample Input-----
3 4
2 1 3
3 1 2 3
2 3 2

-----Sample Output-----
1

As only the third food is liked by all the three people, 1 should be printed.
user: Given is an integer N.
Takahashi chooses an integer a from the positive integers not greater than N with equal probability.
Find the probability that a is odd.

-----Constraints-----
 - 1 \leq N \leq 100

-----Input-----
Input is given from Standard Input in the following format:
N

-----Output-----
Print the probability that a is odd.
Your output will be considered correct when its absolute or relative error from the judge's output is at most 10^{-6}.

-----Sample Input-----
4

-----Sample Output-----
0.5000000000

There are four positive integers not greater than 4: 1, 2, 3, and 4. Among them, we have two odd numbers: 1 and 3. Thus, the answer is \frac{2}{4} = 0.5.
user: There are N people numbered 1 to N. Each of them is either an honest person whose testimonies are always correct or an unkind person whose testimonies may be correct or not.
Person i gives A_i testimonies. The j-th testimony by Person i is represented by two integers x_{ij} and y_{ij}. If y_{ij} = 1, the testimony says Person x_{ij} is honest; if y_{ij} = 0, it says Person x_{ij} is unkind.
How many honest persons can be among those N people at most?

-----Constraints-----
 - All values in input are integers.
 - 1 \leq N \leq 15
 - 0 \leq A_i \leq N - 1
 - 1 \leq x_{ij} \leq N
 - x_{ij} \neq i
 - x_{ij_1} \neq x_{ij_2} (j_1 \neq j_2)
 - y_{ij} = 0, 1

-----Input-----
Input is given from Standard Input in the following format:
N
A_1
x_{11} y_{11}
x_{12} y_{12}
:
x_{1A_1} y_{1A_1}
A_2
x_{21} y_{21}
x_{22} y_{22}
:
x_{2A_2} y_{2A_2}
:
A_N
x_{N1} y_{N1}
x_{N2} y_{N2}
:
x_{NA_N} y_{NA_N}

-----Output-----
Print the maximum possible number of honest persons among the N people.

-----Sample Input-----
3
1
2 1
1
1 1
1
2 0

-----Sample Output-----
2

If Person 1 and Person 2 are honest and Person 3 is unkind, we have two honest persons without inconsistencies, which is the maximum possible number of honest persons.
user: We have a string S consisting of lowercase English letters.
If the length of S is at most K, print S without change.
If the length of S exceeds K, extract the first K characters in S, append ... to the end of them, and print the result.

-----Constraints-----
 - K is an integer between 1 and 100 (inclusive).
 - S is a string consisting of lowercase English letters.
 - The length of S is between 1 and 100 (inclusive).

-----Input-----
Input is given from Standard Input in the following format:
K
S

-----Output-----
Print a string as stated in Problem Statement.

-----Sample Input-----
7
nikoandsolstice

-----Sample Output-----
nikoand...

nikoandsolstice has a length of 15, which exceeds K=7.
We should extract the first 7 characters in this string, append ... to the end of them, and print the result nikoand....
user: In the Kingdom of AtCoder, people use a language called Taknese, which uses lowercase English letters.
In Taknese, the plural form of a noun is spelled based on the following rules:
 - If a noun's singular form does not end with s, append s to the end of the singular form.
 - If a noun's singular form ends with s, append es to the end of the singular form.
You are given the singular form S of a Taknese noun. Output its plural form.

-----Constraints-----
 - S is a string of length 1 between 1000, inclusive.
 - S contains only lowercase English letters.

-----Input-----
Input is given from Standard Input in the following format:
S

-----Output-----
Print the plural form of the given Taknese word.

-----Sample Input-----
apple

-----Sample Output-----
apples

apple ends with e, so its plural form is apples.
user: N Snukes called Snuke 1, Snuke 2, ..., Snuke N live in a town.
There are K kinds of snacks sold in this town, called Snack 1, Snack 2, ..., Snack K. The following d_i Snukes have Snack i: Snuke A_{i, 1}, A_{i, 2}, \cdots, A_{i, {d_i}}.
Takahashi will walk around this town and make mischief on the Snukes who have no snacks. How many Snukes will fall victim to Takahashi's mischief?

-----Constraints-----
 - All values in input are integers.
 - 1 \leq N \leq 100
 - 1 \leq K \leq 100
 - 1 \leq d_i \leq N
 - 1 \leq A_{i, 1} < \cdots < A_{i, d_i} \leq N

-----Input-----
Input is given from Standard Input in the following format:
N K
d_1
A_{1, 1} \cdots A_{1, d_1}
\vdots
d_K
A_{K, 1} \cdots A_{K, d_K}

-----Output-----
Print the answer.

-----Sample Input-----
3 2
2
1 3
1
3

-----Sample Output-----
1

 - Snuke 1 has Snack 1.
 - Snuke 2 has no snacks.
 - Snuke 3 has Snack 1 and 2.
Thus, there will be one victim: Snuke 2.
user: There are N slimes lining up from left to right. The colors of these slimes will be given as a string S of length N consisting of lowercase English letters. The i-th slime from the left has the color that corresponds to the i-th character of S.
Adjacent slimes with the same color will fuse into one larger slime without changing the color. If there were a slime adjacent to this group of slimes before fusion, that slime is now adjacent to the new larger slime.
Ultimately, how many slimes will be there?

-----Constraints-----
 - 1 \leq N \leq 10^5
 - |S| = N
 - S consists of lowercase English letters.

-----Input-----
Input is given from Standard Input in the following format:
N
S

-----Output-----
Print the final number of slimes.

-----Sample Input-----
10
aabbbbaaca

-----Sample Output-----
5

Ultimately, these slimes will fuse into abaca.
user: As AtCoder Beginner Contest 100 is taking place, the office of AtCoder, Inc. is decorated with a sequence of length N, a = {a_1, a_2, a_3, ..., a_N}.

Snuke, an employee, would like to play with this sequence.
Specifically, he would like to repeat the following operation as many times as possible:
For every i satisfying 1 \leq i \leq N, perform one of the following: "divide a_i by 2" and "multiply a_i by 3".  
Here, choosing "multiply a_i by 3" for every i is not allowed, and the value of a_i after the operation must be an integer.

At most how many operations can be performed?

-----Constraints-----
 - N is an integer between 1 and 10 \ 000 (inclusive).
 - a_i is an integer between 1 and 1 \ 000 \ 000 \ 000 (inclusive).

-----Input-----
Input is given from Standard Input in the following format:
N
a_1 a_2 a_3 ... a_N

-----Output-----
Print the maximum number of operations that Snuke can perform.

-----Sample Input-----
3
5 2 4

-----Sample Output-----
3

The sequence is initially {5, 2, 4}. Three operations can be performed as follows:
 - First, multiply a_1 by 3, multiply a_2 by 3 and divide a_3 by 2. The sequence is now {15, 6, 2}.
 - Next, multiply a_1 by 3, divide a_2 by 2 and multiply a_3 by 3. The sequence is now {45, 3, 6}.
 - Finally, multiply a_1 by 3, multiply a_2 by 3 and divide a_3 by 2. The sequence is now {135, 9, 3}.
user: We have A cards, each of which has an integer 1 written on it. Similarly, we also have B cards with 0s and C cards with -1s.
We will pick up K among these cards. What is the maximum possible sum of the numbers written on the cards chosen?

-----Constraints-----
 - All values in input are integers.
 - 0 \leq A, B, C
 - 1 \leq K \leq A + B + C \leq 2 \times 10^9

-----Input-----
Input is given from Standard Input in the following format:
A B C K

-----Output-----
Print the maximum possible sum of the numbers written on the cards chosen.

-----Sample Input-----
2 1 1 3

-----Sample Output-----
2

Consider picking up two cards with 1s and one card with a 0.
In this case, the sum of the numbers written on the cards is 2, which is the maximum possible value.
user: There are some animals in a garden. Each of them is a crane with two legs or a turtle with four legs.
Takahashi says: "there are X animals in total in the garden, and they have Y legs in total." Determine whether there is a combination of numbers of cranes and turtles in which this statement is correct.

-----Constraints-----
 - 1 \leq X \leq 100
 - 1 \leq Y \leq 100
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
X Y

-----Output-----
If there is a combination of numbers of cranes and turtles in which the statement is correct, print Yes; otherwise, print No.

-----Sample Input-----
3 8

-----Sample Output-----
Yes

The statement "there are 3 animals in total in the garden, and they have 8 legs in total" is correct if there are two cranes and one turtle. Thus, there is a combination of numbers of cranes and turtles in which the statement is correct.
user: You are given an undirected unweighted graph with N vertices and M edges that contains neither self-loops nor double edges.

Here, a self-loop is an edge where a_i = b_i (1≤i≤M), and double edges are two edges where (a_i,b_i)=(a_j,b_j) or (a_i,b_i)=(b_j,a_j) (1≤i<j≤M).

How many different paths start from vertex 1 and visit all the vertices exactly once?

Here, the endpoints of a path are considered visited.  
For example, let us assume that the following undirected graph shown in Figure 1 is given.
Figure 1: an example of an undirected graph
The following path shown in Figure 2 satisfies the condition.
Figure 2: an example of a path that satisfies the condition
However, the following path shown in Figure 3 does not satisfy the condition, because it does not visit all the vertices.
Figure 3: an example of a path that does not satisfy the condition
Neither the following path shown in Figure 4, because it does not start from vertex 1.
Figure 4: another example of a path that does not satisfy the condition

-----Constraints-----
 - 2≦N≦8 
 - 0≦M≦N(N-1)/2 
 - 1≦a_i<b_i≦N 
 - The given graph contains neither self-loops nor double edges.

-----Input-----
The input is given from Standard Input in the following format:
N M  
a_1 b_1  
a_2 b_2
:  
a_M b_M  

-----Output-----
Print the number of the different paths that start from vertex 1 and visit all the vertices exactly once.

-----Sample Input-----
3 3
1 2
1 3
2 3

-----Sample Output-----
2

The given graph is shown in the following figure:
The following two paths satisfy the condition:
user: You have N apples, called Apple 1, Apple 2, Apple 3, ..., Apple N. The flavor of Apple i is L+i-1, which can be negative.
You can make an apple pie using one or more of the apples. The flavor of the apple pie will be the sum of the flavors of the apples used.
You planned to make an apple pie using all of the apples, but being hungry tempts you to eat one of them, which can no longer be used to make the apple pie.
You want to make an apple pie that is as similar as possible to the one that you planned to make. Thus, you will choose the apple to eat so that the flavor of the apple pie made of the remaining N-1 apples will have the smallest possible absolute difference from the flavor of the apple pie made of all the N apples.
Find the flavor of the apple pie made of the remaining N-1 apples when you choose the apple to eat as above.
We can prove that this value is uniquely determined.

-----Constraints-----
 - 2 \leq N \leq 200
 - -100 \leq L \leq 100
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
N L

-----Output-----
Find the flavor of the apple pie made of the remaining N-1 apples when you optimally choose the apple to eat.

-----Sample Input-----
5 2

-----Sample Output-----
18

The flavors of Apple 1, 2, 3, 4, and 5 are 2, 3, 4, 5, and 6, respectively. The optimal choice is to eat Apple 1, so the answer is 3+4+5+6=18.
user: Let us define the FizzBuzz sequence a_1,a_2,... as follows:
 - If both 3 and 5 divides i, a_i=\mbox{FizzBuzz}.
 - If the above does not hold but 3 divides i, a_i=\mbox{Fizz}.
 - If none of the above holds but 5 divides i, a_i=\mbox{Buzz}.
 - If none of the above holds, a_i=i.
Find the sum of all numbers among the first N terms of the FizzBuzz sequence.

-----Constraints-----
 - 1 \leq N \leq 10^6

-----Input-----
Input is given from Standard Input in the following format:
N

-----Output-----
Print the sum of all numbers among the first N terms of the FizzBuzz sequence.

-----Sample Input-----
15

-----Sample Output-----
60

The first 15 terms of the FizzBuzz sequence are:
1,2,\mbox{Fizz},4,\mbox{Buzz},\mbox{Fizz},7,8,\mbox{Fizz},\mbox{Buzz},11,\mbox{Fizz},13,14,\mbox{FizzBuzz}
Among them, numbers are 1,2,4,7,8,11,13,14, and the sum of them is 60.
user: Given are an integer X and an integer sequence of length N: p_1, \ldots, p_N.
Among the integers not contained in the sequence p_1, \ldots, p_N (not necessarily positive), find the integer nearest to X, that is, find the integer whose absolute difference with X is the minimum. If there are multiple such integers, report the smallest such integer.

-----Constraints-----
 - 1 \leq X \leq 100
 - 0 \leq N \leq 100
 - 1 \leq p_i \leq 100
 - p_1, \ldots, p_N are all distinct.
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
X N
p_1 ... p_N

-----Output-----
Print the answer.

-----Sample Input-----
6 5
4 7 10 6 5

-----Sample Output-----
8

Among the integers not contained in the sequence 4, 7, 10, 6, 5, the one nearest to 6 is 8.
user: There are H rows and W columns of white square cells.
You will choose h of the rows and w of the columns, and paint all of the cells contained in those rows or columns.
How many white cells will remain?
It can be proved that this count does not depend on what rows and columns are chosen.

-----Constraints-----
 - All values in input are integers.
 - 1 \leq H, W \leq 20
 - 1 \leq h \leq H
 - 1 \leq w \leq W

-----Input-----
Input is given from Standard Input in the following format:
H W
h w

-----Output-----
Print the number of white cells that will remain.

-----Sample Input-----
3 2
2 1

-----Sample Output-----
1

There are 3 rows and 2 columns of cells. When two rows and one column are chosen and painted in black, there is always one white cell that remains.
user: You are given a sequence of integers of length $n$ and integer number $k$. You should print any integer number $x$ in the range of $[1; 10^9]$ (i.e. $1 \le x \le 10^9$) such that exactly $k$ elements of given sequence are less than or equal to $x$.

Note that the sequence can contain equal elements.

If there is no such $x$, print "-1" (without quotes).


-----Input-----

The first line of the input contains integer numbers $n$ and $k$ ($1 \le n \le 2 \cdot 10^5$, $0 \le k \le n$). The second line of the input contains $n$ integer numbers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$) — the sequence itself.


-----Output-----

Print any integer number $x$ from range $[1; 10^9]$ such that exactly $k$ elements of given sequence is less or equal to $x$.

If there is no such $x$, print "-1" (without quotes).


-----Examples-----
Input
7 4
3 7 5 1 10 3 20

Output
6
Input
7 2
3 7 5 1 10 3 20

Output
-1



-----Note-----

In the first example $5$ is also a valid answer because the elements with indices $[1, 3, 4, 6]$ is less than or equal to $5$ and obviously less than or equal to $6$.

In the second example you cannot choose any number that only $2$ elements of the given sequence will be less than or equal to this number because $3$ elements of the given sequence will be also less than or equal to this number.