Problem Statement

You are given N strings.

The i-th string S_i (1 \leq i \leq N) is either Takahashi or Aoki.

How many i are there such that S_i is equal to Takahashi?

Constraints

• 1 \leq N \leq 100
• N is an integer.
• Each S_i is Takahashi or Aoki. (1 \leq i \leq N)

Sample Input 1

3
Aoki
Takahashi
Takahashi

Sample Output 1

2

S_2 and S_3 are equal to Takahashi, while S_1 is not.

Therefore, print 2.

Sample Input 2

2
Aoki
Aoki

Sample Output 2

0

It is possible that no S_i is equal to Takahashi.

Sample Input 3

20
Aoki
Takahashi
Takahashi
Aoki
Aoki
Aoki
Aoki
Takahashi
Aoki
Aoki
Aoki
Takahashi
Takahashi
Aoki
Takahashi
Aoki
Aoki
Aoki
Aoki
Takahashi

Sample Output 3

7

Problem Statement

There are 2N people standing in a row, and the person at the i-th position from the left is wearing clothes of color A_i. Here, the clothes have N colors from 1 to N, and exactly two people are wearing clothes of each color.

Find how many of the integers i=1,2,\ldots,N satisfy the following condition:

• There is exactly one person between the two people wearing clothes of color i.

Constraints

• 2 \leq N \leq 100
• 1 \leq A_i \leq N
• Each integer from 1 through N appears exactly twice in A.
• All input values are integers.

Sample Input 1

3
1 2 1 3 2 3

Sample Output 1

2

There are two values of i that satisfy the condition: 1 and 3.

In fact, the people wearing clothes of color 1 are at the 1st and 3rd positions from the left, with exactly one person in between.

Sample Input 2

2
1 1 2 2

Sample Output 2

0

There may be no i that satisfies the condition.

Sample Input 3

4
4 3 2 3 2 1 4 1

Sample Output 3

3

Problem Statement

The coordinate plane is covered with 2\times1 tiles. The tiles are laid out according to the following rules:

• For an integer pair (i,j), the square A _ {i,j}=\lbrace(x,y)\mid i\leq x\leq i+1\wedge j\leq y\leq j+1\rbrace is contained in one tile.
• When i+j is even, A _ {i,j} and A _ {i + 1,j} are contained in the same tile.

Tiles include their boundaries, and no two different tiles share a positive area.

Near the origin, the tiles are laid out as follows:

Takahashi starts at the point (S _ x+0.5,S _ y+0.5) on the coordinate plane.

He can repeat the following move as many times as he likes:

• Choose a direction (up, down, left, or right) and a positive integer n. Move n units in that direction.

Each time he enters a tile, he pays a toll of 1.

Find the minimum toll he must pay to reach the point (T _ x+0.5,T _ y+0.5).

Constraints

• 0\leq S _ x\leq2\times10 ^ {16}
• 0\leq S _ y\leq2\times10 ^ {16}
• 0\leq T _ x\leq2\times10 ^ {16}
• 0\leq T _ y\leq2\times10 ^ {16}
• All input values are integers.

Sample Input 1

5 0
2 5

Sample Output 1

5

For example, Takahashi can pay a toll of 5 by moving as follows:

• Move left by 1. Pay a toll of 0.
• Move up by 1. Pay a toll of 1.
• Move left by 1. Pay a toll of 0.
• Move up by 3. Pay a toll of 3.
• Move left by 1. Pay a toll of 0.
• Move up by 1. Pay a toll of 1.

It is impossible to reduce the toll to 4 or less, so print 5.

Sample Input 2

3 1
4 1

Sample Output 2

0

There are cases where no toll needs to be paid.

Sample Input 3

2552608206527595 5411232866732612
771856005518028 7206210729152763

Sample Output 3

1794977862420151

Note that the value to be output may exceed the range of a 32-bit integer.

Problem Statement

You are given a string S of length N consisting of characters A, B, and ?.

You are also given a positive integer K. A string T consisting of A and B is considered a good string if it satisfies the following condition:

• No contiguous substring of length K in T is a palindrome.

Let q be the number of ? characters in S. There are 2^q strings that can be obtained by replacing each ? in S with either A or B. Find how many of these strings are good strings.

The count can be very large, so find it modulo 998244353.

Constraints

• 2 \leq K \leq N \leq 1000
• K \leq 10
• S is a string consisting of A, B, and ?.
• The length of S is N.
• N and K are integers.

Sample Input 1

7 4
AB?A?BA

Sample Output 1

1

The given string has two ?s. There are four strings obtained by replacing each ? with A or B:

• ABAAABA
• ABAABBA
• ABBAABA
• ABBABBA

Among these, the last three contain the contiguous substring ABBA of length 4, which is a palindrome, and thus are not good strings.

Therefore, you should print 1.

Sample Input 2

40 7
????????????????????????????????????????

Sample Output 2

116295436

Ensure to find the number of good strings modulo 998244353.

Sample Input 3

15 5
ABABA??????????

Sample Output 3

0

It is possible that there is no way to replace the ?s to obtain a good string.

Sample Input 4

40 8
?A?B??B?B?AA?A?B??B?A???B?BB?B???BA??BAA

Sample Output 4

259240

Story

There is a long water tank with boards of different heights placed at equal intervals. Takahashi wants to know the time at which water reaches each section separated by the boards when water is poured from one end of the tank.

Problem Statement

You are given a sequence of positive integers of length N: H=(H _ 1,H _ 2,\dotsc,H _ N).

There is a sequence of non-negative integers of length N+1: A=(A _ 0,A _ 1,\dotsc,A _ N). Initially, A _ 0=A _ 1=\dotsb=A _ N=0.

Perform the following operations repeatedly on A:

1. Increase the value of A _ 0 by 1.
2. For i=1,2,\ldots,N in this order, perform the following operation:
   • If A _ {i-1}\gt A _ i and A _ {i-1}\gt H _ i, decrease the value of A _ {i-1} by 1 and increase the value of A _ i by 1.

For each i=1,2,\ldots,N, find the number of operations before A _ i>0 holds for the first time.

Constraints

• 1\leq N\leq2\times10 ^ 5
• 1\leq H _ i\leq10 ^ 9\ (1\leq i\leq N)
• All input values are integers.

Sample Input 1

5
3 1 4 1 5

Sample Output 1

4 5 13 14 26

The first five operations go as follows.

Here, each row corresponds to one operation, with the leftmost column representing step 1 and the others representing step 2.

From this diagram, A _ 1\gt0 holds for the first time after the 4th operation, and A _ 2\gt0 holds for the first time after the 5th operation.

Similarly, the answers for A _ 3, A _ 4, A _ 5 are 13, 14, 26, respectively.

Therefore, you should print 4 5 13 14 26.

Sample Input 2

6
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000

Sample Output 2

1000000001 2000000001 3000000001 4000000001 5000000001 6000000001

Note that the values to be output may not fit within a 32-bit integer.

Sample Input 3

15
748 169 586 329 972 529 432 519 408 587 138 249 656 114 632

Sample Output 3

749 918 1921 2250 4861 5390 5822 6428 6836 7796 7934 8294 10109 10223 11373

Problem Statement

You are given a sequence of integers A=(A_1,\ldots,A_N). For a tree T with N vertices, define f(T) as follows:

• Let d_i be the degree of vertex i in T. Then, f(T)=\sum_{i=1}^N {d_i}^2 A_i.

Find the minimum possible value of f(T).

The constraints guarantee the answer to be less than 2^{63}.

Constraints

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

Sample Input 1

4
3 2 5 2

Sample Output 1

24

Consider a tree T with an edge connecting vertices 1 and 2, an edge connecting vertices 2 and 4, and an edge connecting vertices 4 and 3.

Then, f(T) = 1^2\times 3 + 2^2\times 2 + 1^2\times 5 + 2^2\times 2 = 24. It can be proven that this is the minimum value of f(T).

Sample Input 2

3
4 3 2

Sample Output 2

15

Sample Input 3

7
10 5 10 2 10 13 15

Sample Output 3

128

Problem Statement

You are given a tree with N vertices. The i-th edge connects vertices u_i and v_i bidirectionally.

Additionally, you are given an integer sequence A=(A_1,\ldots,A_N).

Here, define f(i,j) as follows:

• If A_i = A_j, then f(i,j) is the minimum number of edges you need to traverse to move from vertex i to vertex j. If A_i \neq A_j, then f(i,j) = 0.

Calculate the value of the following expression:

Constraints

• 2 \leq N \leq 2 \times 10^5
• 1 \leq u_i, v_i \leq N
• 1 \leq A_i \leq N
• The input graph is a tree.
• All input values are integers.

Sample Input 1

4
3 4
4 2
1 2
2 1 1 2

Sample Output 1

4

f(1,4)=2, f(2,3)=2. For all other i, j\ (1 \leq i < j \leq N), we have f(i,j)=0, so the answer is 2+2=4.

Sample Input 2

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

Sample Output 2

19