Problem Statement

One day, tired from going to school, Takahashi wanted to know how many days there were until Saturday.
We know that the day was a weekday, and the name of the day of the week was S in English.
How many days were there until the first Saturday after that day (including Saturday but not the starting day)?

Constraints

• S is Monday, Tuesday, Wednesday, Thursday, or Friday.

Sample Input 1

Wednesday

Sample Output 1

3

It was Wednesday, so there were 3 days until the first Saturday after that day.

Sample Input 2

Monday

Sample Output 2

5

Problem Statement

You are given a string S consisting of A, B, and C. S is guaranteed to contain all of A, B, and C.

If the characters of S are checked one by one from the left, how many characters will have been checked when the following condition is satisfied for the first time?

• All of A, B, and C have appeared at least once.

Constraints

• 3 \leq N \leq 100
• S is a string of length N consisting of A, B, and C.
• S contains all of A, B, and C.

Sample Input 1

5
ACABB

Sample Output 1

4

In the first four characters from the left, A, B, and C appear twice, once, and once, respectively, satisfying the condition.
The condition is not satisfied by checking three or fewer characters, so the answer is 4.

Sample Input 2

4
CABC

Sample Output 2

3

In the first three characters from the left, each of A, B, and C appears once, satisfying the condition.

Sample Input 3

30
AABABBBABABBABABCABACAABCBACCA

Sample Output 3

17

Problem Statement

There are N platforms arranged in a row. The height of the i-th platform from the left is H_i.

Takahashi is initially standing on the leftmost platform.

Since he likes heights, he will repeat the following move as long as possible.

• If the platform he is standing on is not the rightmost one, and the next platform to the right has a height greater than that of the current platform, step onto the next platform.

Find the height of the final platform he will stand on.

Constraints

• 2 \leq N \leq 10^5
• 1 \leq H_i \leq 10^9
• All values in input are integers.

Sample Input 1

5
1 5 10 4 2

Sample Output 1

10

Takahashi is initially standing on the leftmost platform, whose height is 1. The next platform to the right has a height of 5 and is higher than the current platform, so he steps onto it.

He is now standing on the 2-nd platform from the left, whose height is 5. The next platform to the right has a height of 10 and is higher than the current platform, so he steps onto it.

He is now standing on the 3-rd platform from the left, whose height is 10. The next platform to the right has a height of 4 and is lower than the current platform, so he stops moving.

Thus, the height of the final platform Takahashi will stand on is 10.

Sample Input 2

3
100 1000 100000

Sample Output 2

100000

Sample Input 3

4
27 1828 1828 9242

Sample Output 3

1828

Problem Statement

N participants, numbered 1 to N, will participate in a contest with M problems, numbered 1 to M.

For an integer i between 1 and N and an integer j between 1 and M, participant i can solve problem j if the j-th character of S_i is o, and cannot solve it if that character is x.

The participants must be in pairs. Print the number of ways to form a pair of participants who can collectively solve all the M problems.

More formally, print the number of pairs (x,y) of integers satisfying 1\leq x < y\leq N such that for any integer j between 1 and M, at least one of participant x and participant y can solve problem j.

Constraints

• N is an integer between 2 and 30, inclusive.
• M is an integer between 1 and 30, inclusive.
• S_i is a string of length M consisting of o and x.

Sample Input 1

5 5
ooooo
oooxx
xxooo
oxoxo
xxxxx

Sample Output 1

5

The following five pairs satisfy the condition: participants 1 and 2, participants 1 and 3, participants 1 and 4, participants 1 and 5, and participants 2 and 3.

On the other hand, the pair of participants 2 and 4, for instance, does not satisfy the condition because they cannot solve problem 4.

Sample Input 2

3 2
ox
xo
xx

Sample Output 2

1

Sample Input 3

2 4
xxxx
oxox

Sample Output 3

0

Problem Statement

We have N cups of ice cream.
The flavor and deliciousness of the i-th cup are F_i and S_i, respectively (S_i is an even number).

You will choose and eat two of the N cups.
Your satisfaction here is defined as follows.

• Let s and t (s \ge t) be the deliciousness of the eaten cups.
  • If the two cups have different flavors, your satisfaction is \displaystyle s+t.
  • Otherwise, your satisfaction is \displaystyle s + \frac{t}{2}.

Find the maximum achievable satisfaction.

Constraints

• All input values are integers.
• 2 \le N \le 3 \times 10^5
• 1 \le F_i \le N
• 2 \le S_i \le 10^9
• S_i is even.

Sample Input 1

4
1 4
2 10
2 8
3 6

Sample Output 1

16

Consider eating the second and fourth cups.

• The second cup has a flavor of 2 and deliciousness of 10.
• The fourth cup has a flavor of 3 and deliciousness of 6.
• Since they have different flavors, your satisfaction is 10+6=16.

Thus, you can achieve the satisfaction of 16.
You cannot achieve a satisfaction greater than 16.

Sample Input 2

4
4 10
3 2
2 4
4 12

Sample Output 2

17

Consider eating the first and fourth cups.

• The first cup has a flavor of 4 and deliciousness of 10.
• The fourth cup has a flavor of 4 and deliciousness of 12.
• Since they have the same flavor, your satisfaction is 12+\frac{10}{2}=17.

Thus, you can achieve the satisfaction of 17.
You cannot achieve a satisfaction greater than 17.

Problem Statement

There are M sets, called S_1, S_2, \dots, S_M, consisting of integers between 1 and N.
S_i consists of C_i integers a_{i, 1}, a_{i, 2}, \dots, a_{i, C_i}.

There are (2^M-1) ways to choose one or more sets from the M sets.
How many of them satisfy the following condition?

• For all integers x such that 1 \leq x \leq N, there is at least one chosen set containing x.

Constraints

• 1 \leq N \leq 10
• 1 \leq M \leq 10
• 1 \leq C_i \leq N
• 1 \leq a_{i,1} \lt a_{i,2} \lt \dots \lt a_{i,C_i} \leq N
• All values in the input are integers.

Sample Input 1

3 3
2
1 2
2
1 3
1
2

Sample Output 1

3

The sets given in the input are S_1 = \lbrace 1, 2 \rbrace, S_2 = \lbrace 1, 3 \rbrace, S_3 = \lbrace 2 \rbrace.
The following three ways satisfy the conditions in the Problem Statement:

• choosing S_1, S_2;
• choosing S_1, S_2, S_3;
• choosing S_2, S_3.

Sample Input 2

4 2
2
1 2
2
1 3

Sample Output 2

0

There may be no way to choose sets that satisfies the conditions in the Problem Statement.

Sample Input 3

6 6
3
2 3 6
3
2 4 6
2
3 6
3
1 5 6
3
1 3 6
2
1 4

Sample Output 3

18

Problem Statement

You are given positive integers N and M.
Find the smallest positive integer X that satisfies both of the conditions below, or print -1 if there is no such integer.

• X can be represented as the product of two integers a and b between 1 and N, inclusive. Here, a and b may be the same.
• X is at least M.

Constraints

• 1\leq N\leq 10^{12}
• 1\leq M\leq 10^{12}
• N and M are integers.

Sample Input 1

5 7

Sample Output 1

8

First, 7 cannot be represented as the product of two integers between 1 and 5.
Second, 8 can be represented as the product of two integers between 1 and 5, such as 8=2\times 4.

Thus, you should print 8.

Sample Input 2

2 5

Sample Output 2

-1

Since 1\times 1=1, 1\times 2=2, 2\times 1=2, and 2\times 2=4, only 1, 2, and 4 can be represented as the product of two integers between 1 and 2, so no number greater than or equal to 5 can be represented as the product of two such integers.
Thus, you should print -1.

Sample Input 3

100000 10000000000

Sample Output 3

10000000000

For a=b=100000 (=10^5), the product of a and b is 10000000000 (=10^{10}), which is the answer.
Note that the answer may not fit into a 32-bit integer type.

Problem Statement

You are given a sequence of positive integers of length N: A=(A_1,A_2,\ldots,A_N). Find the number of triples of positive integers (i,j,k) that satisfy all of the following conditions:

• 1\leq i < j < k\leq N,
• A_i = A_k,
• A_i \neq A_j.

Constraints

• 3\leq N\leq 3\times 10^5
• 1\leq A_i \leq N
• All input values are integers.

Sample Input 1

5
1 2 1 3 2

Sample Output 1

3

The following three triples of positive integers (i,j,k) satisfy the conditions:

• (i,j,k)=(1,2,3)
• (i,j,k)=(2,3,5)
• (i,j,k)=(2,4,5)

Sample Input 2

7
1 2 3 4 5 6 7

Sample Output 2

0

There may be no triples of positive integers (i,j,k) that satisfy the conditions.

Sample Input 3

13
9 7 11 7 3 8 1 13 11 11 11 6 13

Sample Output 3

20

Problem Statement

There are N cities numbered city 1, city 2, \ldots, and city N.
There are also one-way teleporters that send you to different cities. Whether a teleporter can send you directly from city i (1\leq i\leq N) to another is represented by a length-M string S_i consisting of 0 and 1. Specifically, for 1\leq j\leq N,

• if 1\leq j-i\leq M and the (j-i)-th character of S_i is 1, then a teleporter can send you directly from city i to city j;
• otherwise, it cannot send you directly from city i to city j.

Solve the following problem for k=2,3,\ldots, N-1:

Can you travel from city 1 to city N without visiting city k by repeatedly using a teleporter? If you can, print the minimum number of times you need to use a teleporter; otherwise, print -1.

Constraints

• 3 \leq N \leq 10^5
• 1\leq M\leq 10
• M<N
• S_i is a string of length M consisting of 0 and 1.
• If i+j>N, then the j-th character of S_i is 0.
• N and M are integers.

Sample Input 1

5 2
11
01
11
10
00

Sample Output 1

2 3 2

A teleporter sends you

• from city 1 to cities 2 and 3;
• from city 2 to city 4;
• from city 3 to cities 4 and 5;
• from city 4 to city 5; and
• from city 5 to nowhere.

Therefore, there are three paths to travel from city 1 to city 5:

• path 1 : city 1 \to city 2 \to city 4 \to city 5;
• path 2 : city 1 \to city 3 \to city 4 \to city 5; and
• path 3 : city 1 \to city 3 \to city 5.

Among these paths,

• two paths, path 2 and path 3, do not visit city 2. Among them, path 3 requires the minimum number of teleporter uses (twice).
• Path 1 is the only path without city 3. It requires using a teleporter three times.
• Path 3 is the only path without city 4. It requires using a teleporter twice.

Thus, 2, 3, and 2, separated by spaces, should be printed.

Sample Input 2

6 3
101
001
101
000
100
000

Sample Output 2

-1 3 3 -1

The only path from city 1 to city 6 is city 1 \to city 2 \to city 5 \to city 6.
For k=2,5, there is no way to travel from city 1 to city 6 without visiting city k.
For k=3,4, the path above satisfies the condition; it requires using a teleporter three times.

Thus, -1, 3, 3, and -1, separated by spaces, should be printed.

Note that a teleporter is one-way; a teleporter can send you from city 3 to city 4, but not from city 4 to city 3,
so the following path, for example, is invalid: city 1 \to city 4 \to city 3 \to city 6.