Problem Statement

You are given a string S of length 6. It is guaranteed that the first three characters of S are ABC and the last three characters are digits.

Determine if S is the abbreviation of a contest held and concluded on AtCoder before the start of this contest.

Here, a string T is "the abbreviation of a contest held and concluded on AtCoder before the start of this contest" if and only if it equals one of the following 348 strings:

ABC001, ABC002, \ldots, ABC314, ABC315, ABC317, ABC318, \ldots, ABC348, ABC349.

Note that ABC316 is not included.

Constraints

• S is a string of length 6 where the first three characters are ABC and the last three characters are digits.

Sample Input 1

ABC349

Sample Output 1

Yes

ABC349 is the abbreviation of a contest held and concluded on AtCoder last week.

Sample Input 2

ABC350

Sample Output 2

No

ABC350 is this contest, which has not concluded yet.

Sample Input 3

ABC316

Sample Output 3

No

ABC316 was not held on AtCoder.

Problem Statement

Two characters x and y are called similar characters if and only if one of the following conditions is satisfied:

• x and y are the same character.
• One of x and y is 1 and the other is l.
• One of x and y is 0 and the other is o.

Two strings S and T, each of length N, are called similar strings if and only if:

• for all i\ (1\leq i\leq N), the i-th character of S and the i-th character of T are similar characters.

Given two length-N strings S and T consisting of lowercase English letters and digits, determine if S and T are similar strings.

Constraints

• N is an integer between 1 and 100.
• Each of S and T is a string of length N consisting of lowercase English letters and digits.

Sample Input 1

3
l0w
1ow

Sample Output 1

Yes

The 1-st character of S is l, and the 1-st character of T is 1. These are similar characters.

The 2-nd character of S is 0, and the 2-nd character of T is o. These are similar characters.

The 3-rd character of S is w, and the 3-rd character of T is w. These are similar characters.

Thus, S and T are similar strings.

Sample Input 2

3
abc
arc

Sample Output 2

No

The 2-nd character of S is b, and the 2-nd character of T is r. These are not similar characters.

Thus, S and T are not similar strings.

Sample Input 3

4
nok0
n0ko

Sample Output 3

Yes

Problem Statement

From the point (0,0) in a two-dimensional plane, let us move the distance of 1 toward the point (A, B). Find our coordinates after the move.

Here, after moving the distance of d from a point X to a point Y (d \le length of the segment XY), we are at the point on the segment XY whose distance from X is d.
The Constraints guarantee that the distance between the points (0, 0) and (A, B) is at least 1.

Constraints

• All values in input are integers.
• 0 \le A,B \le 1000
• (A,B) \neq (0,0)

Sample Input 1

3 4

Sample Output 1

0.600000000000 0.800000000000

Printing 0.5999999999 0.8000000001, for example, would also be accepted.

Sample Input 2

1 0

Sample Output 2

1.000000000000 0.000000000000

We may arrive at (A, B).

Sample Input 3

246 402

Sample Output 3

0.521964870245 0.852966983083

Problem Statement

A supermarket sells egg packs.

A pack of 6 eggs costs S yen, a pack of 8 eggs costs M yen, and a pack of 12 eggs costs L yen.

When you can buy any number of each pack, find the minimum amount of money required to purchase at least N eggs.

Constraints

• 1 \leq N \leq 100
• 1 \leq S,M,L \leq 10^4
• All input values are integers.

Sample Input 1

16 120 150 200

Sample Output 1

300

It is optimal to buy two 8-egg packs.

Sample Input 2

10 100 50 10

Sample Output 2

10

It is optimal to buy one 12-egg pack.

Sample Input 3

99 600 800 1200

Sample Output 3

10000

It is optimal to buy five 8-egg packs and five 12-egg packs.

Problem Statement

We have N bags.
Bag i contains L_i balls. The j-th ball (1\leq j\leq L_i) in Bag i has a positive integer a_{i,j} written on it.

We will pick out one ball from each bag.
How many ways are there to pick the balls so that the product of the numbers written on the picked balls is X?

Here, we distinguish all balls, even with the same numbers written on them.

Constraints

• N \geq 2
• L_i \geq 2
• The product of the numbers of balls in the bags is at most 10^5: \displaystyle\prod_{i=1}^{N}L_i \leq 10^5.
• 1 \leq a_{i,j} \leq 10^9
• 1 \leq X \leq 10^{18}
• All values in input are integers.

Sample Input 1

2 40
3 1 8 4
2 10 5

Sample Output 1

2

When choosing the 3-rd ball in Bag 1 and 1-st ball in Bag 2, we have a_{1,3} \times a_{2,1} = 4 \times 10 = 40.
When choosing the 2-nd ball in Bag 1 and 2-nd ball in Bag 2, we have a_{1,2} \times a_{2,2} = 8 \times 5 = 40.
There are no other ways to make the product 40, so the answer is 2.

Sample Input 2

3 200
3 10 10 10
3 10 10 10
5 2 2 2 2 2

Sample Output 2

45

Note that we distinguish all balls, even with the same numbers written on them.

Sample Input 3

3 1000000000000000000
2 1000000000 1000000000
2 1000000000 1000000000
2 1000000000 1000000000

Sample Output 3

0

There may be no way to make the product X.

Problem Statement

You are given an integer N. Consider permuting the digits in N and separate them into two positive integers.

For example, for the integer 123, there are six ways to separate it, as follows:

• 12 and 3,
• 21 and 3,
• 13 and 2,
• 31 and 2,
• 23 and 1,
• 32 and 1.

Here, the two integers after separation must not contain leading zeros. For example, it is not allowed to separate the integer 101 into 1 and 01. Additionally, since the resulting integers must be positive, it is not allowed to separate 101 into 11 and 0, either.

What is the maximum possible product of the two resulting integers, obtained by the optimal separation?

Constraints

• N is an integer between 1 and 10^9 (inclusive).
• N contains two or more digits that are not 0.

Sample Input 1

123

Sample Output 1

63

As described in Problem Statement, there are six ways to separate it:

• 12 and 3,
• 21 and 3,
• 13 and 2,
• 31 and 2,
• 23 and 1,
• 32 and 1.

The products of these pairs, in this order, are 36, 63, 26, 62, 23, 32, with 63 being the maximum.

Sample Input 2

1010

Sample Output 2

100

There are two ways to separate it:

• 100 and 1,
• 10 and 10.

In either case, the product is 100.

Sample Input 3

998244353

Sample Output 3

939337176

Problem Statement

Takahashi found a puzzle along some road.
It is composed of an undirected graph with nine vertices and M edges, and eight pieces.

The nine vertices of the graph are called Vertex 1, Vertex 2, \ldots, Vertex 9. For each i = 1, 2, \ldots, M, the i-th edge connects Vertex u_i and Vertex v_i.
The eight pieces are called Piece 1, Piece 2, \ldots, Piece 8. For each j = 1, 2, \ldots, 8, Piece j is on Vertex p_j.
Here, it is guaranteed that all pieces are on distinct vertices. Note that there is exactly one empty vertex without a piece.

Takahashi can do the following operation on the puzzle any number of times (possibly zero).

Choose a piece on a vertex adjacent to the empty vertex, and move it to the empty vertex.

By repeating this operation, he aims to complete the puzzle. The puzzle is considered complete when the following holds.

• For each j = 1, 2, \ldots, 8, Piece j is on Vertex j.

Determine whether it is possible for Takahashi to complete the puzzle. If it is possible, find the minimum number of operations needed to do so.

Constraints

• 0 \leq M \leq 36
• 1 \leq u_i, v_i \leq 9
• The given graph has no multi-edges or self-loops.
• 1 \leq p_j \leq 9
• j \neq j' \Rightarrow p_j \neq p_{j'}
• All values in input are integers.

Sample Input 1

5
1 2
1 3
1 9
2 9
3 9
3 9 2 4 5 6 7 8

Sample Output 1

5

The following procedure completes the puzzle in five operations.

1. Move Piece 2 from Vertex 9 to Vertex 1.
2. Move Piece 3 from Vertex 2 to Vertex 9.
3. Move Piece 2 from Vertex 1 to Vertex 2.
4. Move Piece 1 from Vertex 3 to Vertex 1.
5. Move Piece 3 from Vertex 9 to Vertex 3.

On the other hand, it is impossible to complete the puzzle in less than five operations. Thus, we should print 5.
Note that the given graph may not be connected.

Sample Input 2

5
1 2
1 3
1 9
2 9
3 9
1 2 3 4 5 6 7 8

Sample Output 2

0

The puzzle is already complete from the beginning. Thus, the minimum number of operations needed to complete the puzzle is 0.

Sample Input 3

12
8 5
9 6
4 5
4 1
2 5
8 9
2 1
3 6
8 7
6 5
7 4
2 3
1 2 3 4 5 6 8 7

Sample Output 3

-1

No sequence of operations can complete the puzzle, so we should print -1.

Sample Input 4

12
6 5
5 4
4 1
4 7
8 5
2 1
2 5
6 9
3 6
9 8
8 7
3 2
2 3 4 6 1 9 7 8

Sample Output 4

16

Problem Statement

Consider the following procedure of, given a string X consisting of lowercase English alphabets, obtaining a new string:

• Split the string X off at the positions where two different characters are adjacent to each other.
• For each string Y that has been split off, replace Y with a string consisting of the character which Y consists of, followed by the length of Y.
• Concatenate the replaced strings without changing the order.

For example, aaabbcccc is divided into aaa,bb,cccc, which are replaced by a3,b2,c4, respectively, which in turn are concatenated without changing the order, resulting in a3b2c4.If the given string is aaaaaaaaaa , the new string is a10 .

Find the number, modulo P, of strings S of lengths N consisting of lowercase English alphabets, such that the length of T is smaller than that of S, where T is the string obtained by the procedure above against the string S.

Constraints

• 1 \le N \le 3000
• 10^8 \le P \le 10^9
• N and P are integers.
• P is a prime.

Sample Input 1

3 998244353

Sample Output 1

26

Those strings of which the 1-st, 2-nd, and 3-rd characters are all the same satisfy the condition.

For example, aaa becomes a3, which satisfies the condition, while abc becomes a1b1c1, which does not.

Sample Input 2

2 998244353

Sample Output 2

0

Note that if a string is transformed into another string of the same length, such as aa that becomes a2, it does not satisfy the condition.

Sample Input 3

5 998244353

Sample Output 3

2626

Strings like aaabb and aaaaa satisfy the condition.

Sample Input 4

3000 924844033

Sample Output 4

607425699

Find the number of strings satisfying the condition modulo P.

Problem Statement

There are N sequences of length M, denoted as A_1, A_2, \ldots, A_N. The i-th sequence is represented by M integers A_{i,1}, A_{i,2}, \ldots, A_{i,M}.

Two sequences X and Y of length M are said to be similar if and only if the number of indices i (1 \leq i \leq M) such that X_i = Y_i is odd.

Find the number of pairs of integers (i,j) satisfying 1 \leq i < j \leq N such that A_i and A_j are similar.

Constraints

• 1 \leq N \leq 2000
• 1 \leq M \leq 2000
• 1 \leq A_{i,j} \leq 999
• All input values are integers.

Sample Input 1

3 3
1 2 3
1 3 4
2 3 4

Sample Output 1

1

The pair (i,j) = (1,2) satisfies the condition because there is only one index k such that A_{1,k} = A_{2,k}, which is k=1.

The pairs (i,j) = (1,3), (2,3) do not satisfy the condition, making (1,2) the only pair that does.

Sample Input 2

6 5
8 27 27 10 24
27 8 2 4 5
15 27 26 17 24
27 27 27 27 27
27 7 22 11 27
19 27 27 27 27

Sample Output 2

5