Problem Statement

You are given a string S consisting of uppercase and lowercase English letters.
Here, exactly one character of S is uppercase, and the others are all lowercase.
Find the integer x such that the x-th character of S is uppercase.
Here, the initial character of S is considered the 1-st one.

Constraints

• S is a string of length between 2 and 100, inclusive, consisting of uppercase and lowercase English letters.
• S has exactly one uppercase letter.

Sample Input 1

aBc

Sample Output 1

2

The 1-st character of aBc is a, the 2-nd is B, and the 3-rd is c; the 2-nd character is uppercase.
Thus, 2 should be printed.

Sample Input 2

xxxxxxXxxx

Sample Output 2

7

An uppercase letter X occurs as the 7-th character of S=xxxxxxXxxx, so 7 should be printed.

Sample Input 3

Zz

Sample Output 3

1

Problem Statement

Find the X-th character from the beginning of the string that is obtained by concatenating these characters: N copies of A's, N copies of B's, …, and N copies of Z's, in this order.

Constraints

• 1 \leq N \leq 100
• 1 \leq X \leq N\times 26
• All values in input are integers.

Sample Input 1

1 3

Sample Output 1

C

We obtain the string ABCDEFGHIJKLMNOPQRSTUVWXYZ, whose 3-rd character from the beginning is C.

Sample Input 2

2 12

Sample Output 2

F

We obtain the string AABBCCDDEEFFGGHHIIJJKKLLMMNNOOPPQQRRSSTTUUVVWWXXYYZZ, whose 12-th character from the beginning is F.

Problem Statement

There are N people numbered 1 to N.
You are given their schedule for the following D days. The schedule for person i is represented by a string S_i of length D. If the j-th character of S_i is o, person i is free on the j-th day; if it is x, they are occupied that day.

From these D days, consider choosing some consecutive days when all the people are free.
How many days can be chosen at most? If no day can be chosen, report 0.

Constraints

• 1 \leq N \leq 100
• 1 \leq D \leq 100
• N and D are integers.
• S_i is a string of length D consisting of o and x.

Sample Input 1

3 5
xooox
oooxx
oooxo

Sample Output 1

2

All the people are free on the second and third days, so we can choose them.
Choosing these two days will maximize the number of days among all possible choices.

Sample Input 2

3 3
oxo
oxo
oxo

Sample Output 2

1

Note that the chosen days must be consecutive. (All the people are free on the first and third days, so we can choose either of them, but not both.)

Sample Input 3

3 3
oox
oxo
xoo

Sample Output 3

0

Print 0 if no day can be chosen.

Sample Input 4

1 7
ooooooo

Sample Output 4

7

Sample Input 5

5 15
oxooooooooooooo
oxooxooooooooox
oxoooooooooooox
oxxxooooooxooox
oxooooooooxooox

Sample Output 5

5

Problem Statement

N people numbered 1,2,\ldots,N were in M photos. In each of the photos, they stood in a single line. In the i-th photo, the j-th person from the left is person a_{i,j}.

Two people who did not stand next to each other in any of the photos may be in a bad mood.

How many pairs of people may be in a bad mood? Here, we do not distinguish a pair of person x and person y, and a pair of person y and person x.

Constraints

• 2 \leq N \leq 50
• 1 \leq M \leq 50
• 1 \leq a_{i,j} \leq N
• a_{i,1},\ldots,a_{i,N} contain each of 1,\ldots,N exactly once.
• All values in the input are integers.

Sample Input 1

4 2
1 2 3 4
4 3 1 2

Sample Output 1

2

The pair of person 1 and person 4, and the pair of person 2 and person 4, may be in a bad mood.

Sample Input 2

3 3
1 2 3
3 1 2
1 2 3

Sample Output 2

0

Sample Input 3

10 10
4 10 7 2 8 3 9 1 6 5
3 6 2 9 1 8 10 7 4 5
9 3 4 5 7 10 1 8 2 6
7 3 1 8 4 9 5 6 2 10
5 2 1 4 10 7 9 8 3 6
5 8 1 6 9 3 2 4 7 10
8 10 3 4 5 7 2 9 6 1
3 10 2 7 8 5 1 4 9 6
10 6 1 5 4 2 3 8 9 7
4 5 9 1 8 2 7 6 3 10

Sample Output 3

6

Problem Statement

On an xy-coordinate plane, is there a lattice point whose distances from two lattice points (x_1, y_1) and (x_2, y_2) are both \sqrt{5}?

Notes

A point on an xy-coordinate plane whose x and y coordinates are both integers is called a lattice point.
The distance between two points (a, b) and (c, d) is defined to be the Euclidean distance between them, \sqrt{(a - c)^2 + (b-d)^2}.

The following figure illustrates an xy-plane with a black circle at (0, 0) and white circles at the lattice points whose distances from (0, 0) are \sqrt{5}. (The grid shows where either x or y is an integer.)

Constraints

• -10^9 \leq x_1 \leq 10^9
• -10^9 \leq y_1 \leq 10^9
• -10^9 \leq x_2 \leq 10^9
• -10^9 \leq y_2 \leq 10^9
• (x_1, y_1) \neq (x_2, y_2)
• All values in input are integers.

Sample Input 1

0 0 3 3

Sample Output 1

Yes

• The distance between points (2,1) and (x_1, y_1) is \sqrt{(0-2)^2 + (0-1)^2} = \sqrt{5};
• the distance between points (2,1) and (x_2, y_2) is \sqrt{(3-2)^2 + (3-1)^2} = \sqrt{5};
• point (2, 1) is a lattice point,

so point (2, 1) satisfies the condition. Thus, Yes should be printed.
One can also assert in the same way that (1, 2) also satisfies the condition.

Sample Input 2

0 1 2 3

Sample Output 2

No

No lattice point satisfies the condition, so No should be printed.

Sample Input 3

1000000000 1000000000 999999999 999999999

Sample Output 3

Yes

Point (10^9 + 1, 10^9 - 2) and point (10^9 - 2, 10^9 + 1) satisfy the condition.

Problem Statement

You are given a simple undirected graph with N vertices numbered 1 to N and M edges numbered 1 to M. Edge i connects vertex u_i and vertex v_i.
Find the number of connected components in this graph.

Notes

A simple undirected graph is a graph that is simple and has undirected edges.
A graph is simple if and only if it has no self-loop or multi-edge.

A subgraph of a graph is a graph formed from some of the vertices and edges of that graph.
A graph is connected if and only if one can travel between every pair of vertices via edges.
A connected component is a connected subgraph that is not part of any larger connected subgraph.

Constraints

• 1 \leq N \leq 100
• 0 \leq M \leq \frac{N(N - 1)}{2}
• 1 \leq u_i, v_i \leq N
• The given graph is simple.
• All values in the input are integers.

Sample Input 1

5 3
1 2
1 3
4 5

Sample Output 1

2

The given graph contains the following two connected components:

• a subgraph formed from vertices 1, 2, 3, and edges 1, 2;
• a subgraph formed from vertices 4, 5, and edge 3.

Sample Input 2

5 0

Sample Output 2

5

Sample Input 3

4 6
1 2
1 3
1 4
2 3
2 4
3 4

Sample Output 3

1

Problem Statement

Takahashi has N kinds of coins; specifically, for 1\leq i\leq N, he has B_i coins worth A_i yen (the currency in Japan) each.

Determine if Takahashi can pay exactly X yen (without change) with the coins he currently has.

Constraints

• 1\leq N\leq 50
• 1\leq X\leq 10^4
• 1\leq A_i\leq 100
• 1\leq B_i\leq 50
• A_i are pairwise distinct.
• All values in the input are integers.

Sample Input 1

2 19
2 3
5 6

Sample Output 1

Yes

Takahashi has three 2-yen coins and six 5-yen coins. He can use two 2-yen coins and three 5-yen coins to pay exactly 2\times 2+5\times 3=19 yen. Thus, Yes should be printed.

Sample Input 2

2 18
2 3
5 6

Sample Output 2

No

There is no combination of the coins that he can use to pay exactly 18 yen. Thus, No should be printed.

Sample Input 3

3 1001
1 1
2 1
100 10

Sample Output 3

Yes

He need not use all kinds of coins.

Problem Statement

There are N kinds of coins used in the Kingdom of Atcoder: A_1-yen, A_2-yen, \ldots, A_N-yen coins.
Here, 1=A_1 < \ldots < A_N holds, and A_{i+1} is a multiple of A_i for every 1\leq i \leq N-1.

When paying for a product worth X yen using just these coins, what is the minimum total number of coins used in payment and returned as change?

Accurately, when Y is an integer at least X, find the minimum sum of the number of coins needed to represent exactly Y yen and the number of coins needed to represent exactly Y-X yen.

Constraints

• All values in input are integers.
• 1 \leq N \leq 60
• 1=A_1 < \ldots <A_N \leq 10^{18}
• A_{i+1} is a multiple of A_i for every 1\leq i \leq N-1.
• 1\leq X \leq 10^{18}

Sample Input 1

3 87
1 10 100

Sample Output 1

5

If we pay with one 100-yen coin and receive one 10-yen coin and three 1-yen coins as the change, the total number of coins will be 5.

Sample Input 2

2 49
1 7

Sample Output 2

7

It is optimal to pay with seven 7-yen coins.

Sample Input 3

10 123456789012345678
1 100 10000 1000000 100000000 10000000000 1000000000000 100000000000000 10000000000000000 1000000000000000000

Sample Output 3

233

Problem Statement

There are N cards numbered 1 through N. Each face of a card has an integer written on it; card i has A_i on its front and B_i on its back. Initially, all cards are face up.

There are M machines numbered 1 through M. Machine j has two (not necessarily distinct) integers X_j and Y_j between 1 and N. If you power up machine j, it flips card X_j with the probability of \frac{1}{2}, and flips card Y_j with the remaining probability of \frac{1}{2}. This probability is independent for each power-up.

Snuke will perform the following procedure.

1. Choose a set S consisting of integers from 1 through M.
2. For each element in S in ascending order, power up the machine with that number.

Among Snuke's possible choices of S, find the maximum expected value of the sum of the integers written on the face-up sides of the cards after the procedure.

Constraints

• 1\leq N \leq 40
• 1\leq M \leq 10^5
• 1\leq A_i,B_i \leq 10^4
• 1\leq X_j,Y_j \leq N
• All input values are integers.

Sample Input 1

3 1
3 10
10 6
5 2
1 2

Sample Output 1

19.500000

If S is chosen to be an empty set, no machine is powered up, so the expected sum of the integers written on the face-up sides of the cards after the procedure is 3+10+5=18.

If S is chosen to be \lbrace 1 \rbrace, machine 1 is powered up.

• If card X_1 = 1 is flipped, the expected sum of the integers written on the face-up sides of the cards after the procedure is 10+10+5=25.
• If card Y_1 = 2 is flipped, the expected sum of the integers written on the face-up sides of the cards after the procedure is 3+6+5=14.

Thus, the expected value is \frac{25+14}{2} = 19.5.

Therefore, the maximum expected value of the sum of the integers written on the face-up sides of the cards after the procedure is 19.5.

Sample Input 2

1 3
5 100
1 1
1 1
1 1

Sample Output 2

100.000000

Different machines may have the same (X_j,Y_j).

Sample Input 3

8 10
6918 9211
16 1868
3857 8537
3340 8506
6263 7940
1449 4593
5902 1932
310 6991
4 4
8 6
3 5
1 1
4 2
5 6
7 5
3 3
1 5
3 1

Sample Output 3

45945.000000