Problem Statement

A regular pentagon P is shown in the figure below.

Determine whether the length of the line segment connecting points S_1 and S_2 of P equals the length of the line segment connecting points T_1 and T_2.

Constraints

• Each of S_1, S_2, T_1, and T_2 is one of the characters A, B, C, D, and E.
• S_1 \neq S_2
• T_1 \neq T_2

Sample Input 1

AC
EC

Sample Output 1

Yes

The length of the line segment connecting point A and point C of P equals the length of the line segment connecting point E and point C.

Sample Input 2

DA
EA

Sample Output 2

No

The length of the line segment connecting point D and point A of P does not equal the length of the line segment connecting point E and point A.

Sample Input 3

BD
BD

Sample Output 3

Yes

Problem Statement

A string S consisting of lowercase English letters is a good string if and only if it satisfies the following property for all integers i not less than 1:

• There are exactly zero or exactly two different letters that appear exactly i times in S.

Given a string S, determine if it is a good string.

Constraints

• S is a string of lowercase English letters with a length between 1 and 100, inclusive.

Sample Input 1

commencement

Sample Output 1

Yes

For the string commencement, the number of different letters that appear exactly i times is as follows:

• i=1: two letters (o and t)
• i=2: two letters (c and n)
• i=3: two letters (e and m)
• i\geq 4: zero letters

Therefore, commencement satisfies the condition of a good string.

Sample Input 2

banana

Sample Output 2

No

For the string banana, there is only one letter that appears exactly one time, which is b, so it does not satisfy the condition of a good string.

Sample Input 3

ab

Sample Output 3

Yes

Problem Statement

There is a tree T with N vertices. The i-th edge (1\leq i\leq N-1) connects vertex U_i and vertex V_i.

You are given two different vertices X and Y in T. List all vertices along the simple path from vertex X to vertex Y in order, including endpoints.

It can be proved that, for any two different vertices a and b in a tree, there is a unique simple path from a to b.

Constraints

• 1\leq N\leq 2\times 10^5
• 1\leq X,Y\leq N
• X\neq Y
• 1\leq U_i,V_i\leq N
• All values in the input are integers.
• The given graph is a tree.

Sample Input 1

5 2 5
1 2
1 3
3 4
3 5

Sample Output 1

2 1 3 5

The tree T is shown below. The simple path from vertex 2 to vertex 5 is 2 \to 1 \to 3 \to 5.
Thus, 2,1,3,5 should be printed in this order, with spaces in between.

Sample Input 2

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

Sample Output 2

1 2

The tree T is shown below.

Problem Statement

The programming contest World Tour Finals is underway, where N players are participating, and half of the competition time has passed. There are M problems in this contest, and the score A_i of problem i is a multiple of 100 between 500 and 2500, inclusive.

For each i = 1, \ldots, N, you are given a string S_i that indicates which problems player i has already solved. S_i is a string of length M consisting of o and x, where the j-th character of S_i is o if player i has already solved problem j, and x if they have not yet solved it. Here, none of the players have solved all the problems yet.

The total score of player i is calculated as the sum of the scores of the problems they have solved, plus a bonus score of i points.
For each i = 1, \ldots, N, answer the following question.

• At least how many of the problems that player i has not yet solved must player i solve to exceed all other players' current total scores?

Note that under the conditions in this statement and the constraints, it can be proved that player i can exceed all other players' current total scores by solving all the problems, so the answer is always defined.

Constraints

• 2\leq N\leq 100
• 1\leq M\leq 100
• 500\leq A_i\leq 2500
• A_i is a multiple of 100.
• S_i is a string of length M consisting of o and x.
• S_i contains at least one x.
• All numeric values in the input are integers.

Sample Input 1

3 4
1000 500 700 2000
xxxo
ooxx
oxox

Sample Output 1

0
1
1

The players' total scores at the halfway point of the competition time are 2001 points for player 1, 1502 points for player 2, and 1703 points for player 3.

Player 1 is already ahead of all other players' total scores without solving any more problems.

Player 2 can, for example, solve problem 4 to have a total score of 3502 points, which would exceed all other players' total scores.

Player 3 can also, for example, solve problem 4 to have a total score of 3703 points, which would exceed all other players' total scores.

Sample Input 2

5 5
1000 1500 2000 2000 2500
xxxxx
oxxxx
xxxxx
oxxxx
oxxxx

Sample Output 2

1
1
1
1
0

Sample Input 3

7 8
500 500 500 500 500 500 500 500
xxxxxxxx
oxxxxxxx
ooxxxxxx
oooxxxxx
ooooxxxx
oooooxxx
ooooooxx

Sample Output 3

7
6
5
4
3
2
0

Problem Statement

The Republic of AtCoder lies on a Cartesian coordinate plane.
It has N towns, numbered 1,2,\dots,N. Town i is at (x_i, y_i), and no two different towns are at the same coordinates.

There are teleportation spells in the nation. A spell is identified by a pair of integers (a,b), and casting a spell (a, b) at coordinates (x, y) teleports you to (x+a, y+b).

Snuke is a great magician who can learn the spell (a, b) for any pair of integers (a, b) of his choice. The number of spells he can learn is also unlimited.
To be able to travel between the towns using spells, he has decided to learn some number of spells so that it is possible to do the following for every pair of different towns (i, j).

• Choose just one spell among the spells learned. Then, repeatedly use just the chosen spell to get from Town i to Town j.

At least how many spells does Snuke need to learn to achieve the objective above?

Constraints

• 2 \leq N \leq 500
• 0 \leq x_i \leq 10^9 (1 \leq i \leq N)
• 0 \leq y_i \leq 10^9 (1 \leq i \leq N)
• (x_i, y_i) \neq (x_j, y_j) if i \neq j.

Sample Input 1

3
1 2
3 6
7 4

Sample Output 1

6

The figure below illustrates the positions of the towns (along with the coordinates of the four corners).

If Snuke learns the six spells below, he can get from Town i to Town j by using one of the spells once for every pair (i,j) (i \neq j), achieving his objective.

• (2, 4)
• (-2, -4)
• (4, -2)
• (-4, 2)
• (-6, -2)
• (6, 2)

Another option is to learn the six spells below. In this case, he can get from Town i to Town j by using one of the spells twice for every pair (i,j) (i \neq j), achieving his objective.

• (1, 2)
• (-1, -2)
• (2, -1)
• (-2, 1)
• (-3, -1)
• (3, 1)

There is no combination of spells that consists of less than six spells and achieve the objective, so we should print 6.

Sample Input 2

3
1 2
2 2
4 2

Sample Output 2

2

The optimal choice is to learn the two spells below:

• (1, 0)
• (-1, 0)

Sample Input 3

4
0 0
0 1000000000
1000000000 0
1000000000 1000000000

Sample Output 3

8