Problem Statement

In the figure shown in the image below, are the points numbered a and b directly connected by a line segment?

Constraints

• 1 \leq a \lt b \leq 10
• a and b are integers.

Sample Input 1

4 5

Sample Output 1

Yes

In the figure shown in the Problem Statement, the points numbered 4 and 5 are directly connected by a line segment.
Thus, Yes should be printed.

Sample Input 2

3 5

Sample Output 2

No

In the figure shown in the Problem Statement, the points numbered 3 and 5 are not directly connected by a line segment.
Thus, No should be printed.

Sample Input 3

1 10

Sample Output 3

Yes

Problem Statement

You are given an integer N. If N is between -2^{31} and 2^{31}-1 (inclusive), print Yes; otherwise, print No.

Constraints

• -2^{63} \leq N < 2^{63}
• N is an integer.

Sample Input 1

10

Sample Output 1

Yes

10 is between -2^{31} and 2^{31}-1, so Yes should be printed.

Sample Input 2

-9876543210

Sample Output 2

No

-9876543210 is less than -2^{31}, so No should be printed.

Sample Input 3

483597848400000

Sample Output 3

No

483597848400000 is greater than 2^{31}-1, so No should be printed.

Problem Statement

There are N people. The family name and given name of the i-th person (1 \leq i \leq N) are S_i and T_i, respectively.

Determine whether there is a pair of people with the same family and given names. In other words, determine whether there is a pair of integers (i,j) such that 1 \leq i \lt j \leq N, S_i=S_j, and T_i=T_j.

Constraints

• 2 \leq N \leq 1000
• N is an integer.
• Each of S_i and T_i is a string of length between 1 and 10 (inclusive) consisting of English lowercase letters.

Sample Input 1

3
tanaka taro
sato hanako
tanaka taro

Sample Output 1

Yes

The first and third persons have the same family and given names.

Sample Input 2

3
saito ichiro
saito jiro
saito saburo

Sample Output 2

No

No two persons have the same family and given names.

Sample Input 3

4
sypdgidop bkseq
bajsqz hh
ozjekw mcybmtt
qfeysvw dbo

Sample Output 3

No

Problem Statement

There is an infinitely long piano keyboard. Is there a continuous segment within this keyboard that consists of W white keys and B black keys?

Let S be the string formed by infinitely repeating the string wbwbwwbwbwbw.

Is there a substring of S that consists of W occurrences of w and B occurrences of b?

Constraints

• W and B are integers.
• 0\leq W,B \leq 100
• W+B \geq 1

Sample Input 1

3 2

Sample Output 1

Yes

The first 15 characters of S are wbwbwwbwbwbwwbw. You can take the 11-th through 15-th characters to form the string bwwbw, which is a substring consisting of three occurrences of w and two occurrences of b.

Sample Input 2

3 0

Sample Output 2

No

The only string consisting of three occurrences of w and zero occurrences of b is www, which is not a substring of S.

Sample Input 3

92 66

Sample Output 3

Yes

Problem Statement

You are given a length-N sequence of non-negative integers.
When B is a sequence obtained by choosing K elements from A and concatenating them without changing the order, find the maximum possible value of MEX(B).

Here, for a sequence X, we define MEX(X) as the unique non-negative integer m that satisfies the following conditions:

• Every integer i such that 0 \le i < m is contained in X.
• m is not contained in X.

Constraints

• All values in the input are integers.
• 1 \le K \le N \le 3 \times 10^5
• 0 \le A_i \le 10^9

Sample Input 1

7 3
2 0 2 3 2 1 9

Sample Output 1

3

In this input, A=(2,0,2,3,2,1,9), and you obtain the sequence B by picking K=3 elements. For example,

• If the 1-st, 2-nd, and 3-rd elements are chosen, MEX(B)=MEX(2,0,2)=1.
• If the 3-rd, 4-th, and 6-th elements are chosen, MEX(B)=MEX(2,3,1)=0.
• If the 2-nd, 6-th, and 7-th elements are chosen, MEX(B)=MEX(0,1,9)=2.
• If the 2-nd, 3-rd, and 6-th elements are chosen, MEX(B)=MEX(0,2,1)=3.

The maximum possible MEX is 3.

Problem Statement

There are N people in an xy-plane. Person i is at (X_i, Y_i). The positions of all people are different.

We have a string S of length N consisting of L and R.
If S_i = R, Person i is facing right; if S_i = L, Person i is facing left. All people simultaneously start walking in the direction they are facing. Here, right and left correspond to the positive and negative x-direction, respectively.

For example, the figure below shows the movement of people when (X_1, Y_1) = (2, 3), (X_2, Y_2) = (1, 1), (X_3, Y_3) =(4, 1), S = RRL.

We say that there is a collision when two people walking in opposite directions come to the same position. Will there be a collision if all people continue walking indefinitely?

Constraints

• 2 \leq N \leq 2 \times 10^5
• 0 \leq X_i \leq 10^9
• 0 \leq Y_i \leq 10^9
• (X_i, Y_i) \neq (X_j, Y_j) if i \neq j.
• All X_i and Y_i are integers.
• S is a string of length N consisting of L and R.

Sample Input 1

3
2 3
1 1
4 1
RRL

Sample Output 1

Yes

This input corresponds to the example in the Problem Statement.
If all people continue walking, Person 2 and Person 3 will collide. Thus, Yes should be printed.

Sample Input 2

2
1 1
2 1
RR

Sample Output 2

No

Since Person 1 and Person 2 walk in the same direction, they never collide.

Sample Input 3

10
1 3
1 4
0 0
0 2
0 4
3 1
2 4
4 2
4 4
3 3
RLRRRLRLRR

Sample Output 3

Yes

Problem Statement

There is a grid with N rows and N columns, where N is an odd number at most 45.

Let (i,j) denote the cell at the i-th row from the top and j-th column from the left.

In this grid, you will place Takahashi and a dragon consisting of N^2-1 parts numbered 1 to N^2-1 in such a way that satisfies the following conditions:

• Takahashi must be placed at the center of the grid, that is, in cell (\frac{N+1}{2},\frac{N+1}{2}).
• Except for the cell where Takahashi is, exactly one dragon part must be placed in each cell.
• For every integer x satisfying 2 \leq x \leq N^2-1, the dragon part x must be placed in a cell adjacent by an edge to the cell containing part x-1.
  • Cells (i,j) and (k,l) are said to be adjacent by an edge if and only if |i-k|+|j-l|=1.

Print one way to arrange the parts to satisfy the conditions. It is guaranteed that there is at least one arrangement that satisfies the conditions.

Constraints

• 3 \leq N \leq 45
• N is odd.

Sample Input 1

5

Sample Output 1

1 2 3 4 5
16 17 18 19 6
15 24 T 20 7
14 23 22 21 8
13 12 11 10 9

The following output also satisfies all the conditions and is correct.

9 10 11 14 15
8 7 12 13 16
5 6 T 18 17
4 3 24 19 20 
1 2 23 22 21

On the other hand, the following outputs are incorrect for the reasons given.

Takahashi is not at the center.

1 2 3 4 5
10 9 8 7 6
11 12 13 14 15
20 19 18 17 16
21 22 23 24 T

The cells containing parts 23 and 24 are not adjacent by an edge.

1 2 3 4 5
10 9 8 7 6
11 12 24 22 23
14 13 T 21 20
15 16 17 18 19

Problem Statement

There are N rectangular cuboids in a three-dimensional space.

These cuboids do not overlap. Formally, for any two different cuboids among them, their intersection has a volume of 0.

The diagonal of the i-th cuboid is a segment that connects two points (X_{i,1},Y_{i,1},Z_{i,1}) and (X_{i,2},Y_{i,2},Z_{i,2}), and its edges are all parallel to one of the coordinate axes.

For each cuboid, find the number of other cuboids that share a face with it.
Formally, for each i, find the number of j with 1\leq j \leq N and j\neq i such that the intersection of the surfaces of the i-th and j-th cuboids has a positive area.

Constraints

• 1 \leq N \leq 10^5
• 0 \leq X_{i,1} < X_{i,2} \leq 100
• 0 \leq Y_{i,1} < Y_{i,2} \leq 100
• 0 \leq Z_{i,1} < Z_{i,2} \leq 100
• Cuboids do not have an intersection with a positive volume.
• All input values are integers.

Sample Input 1

4
0 0 0 1 1 1
0 0 1 1 1 2
1 1 1 2 2 2
3 3 3 4 4 4

Sample Output 1

1
1
0
0

The 1-st and 2-nd cuboids share a rectangle whose diagonal is the segment connecting two points (0,0,1) and (1,1,1).
The 1-st and 3-rd cuboids share a point (1,1,1), but do not share a surface.

Sample Input 2

3
0 0 10 10 10 20
3 4 1 15 6 10
0 9 6 1 20 10

Sample Output 2

2
1
1

Sample Input 3

8
0 0 0 1 1 1
0 0 1 1 1 2
0 1 0 1 2 1
0 1 1 1 2 2
1 0 0 2 1 1
1 0 1 2 1 2
1 1 0 2 2 1
1 1 1 2 2 2

Sample Output 3

3
3
3
3
3
3
3
3

Problem Statement

You are given a simple connected undirected graph with N vertices and M edges. (A graph is said to be simple if it has no multi-edges and no self-loops.)
For i = 1, 2, \ldots, M, the i-th edge connects Vertex u_i and Vertex v_i.

A sequence (A_1, A_2, \ldots, A_k) is said to be a path of length k if both of the following two conditions are satisfied:

• For all i = 1, 2, \dots, k, it holds that 1 \leq A_i \leq N.
• For all i = 1, 2, \ldots, k-1, Vertex A_i and Vertex A_{i+1} are directly connected by an edge.

An empty sequence is regarded as a path of length 0.

Let S = s_1s_2\ldots s_N be a string of length N consisting of 0 and 1. A path A = (A_1, A_2, \ldots, A_k) is said to be a good path with respect to S if the following conditions are satisfied:

• For all i = 1, 2, \ldots, N, it holds that:
  • if s_i = 0, then A has even number of i's.
  • if s_i = 1, then A has odd number of i's.

There are 2^N possible S (in other words, there are 2^N strings of length N consisting of 0 and 1). Find the sum of "the length of the shortest good path with respect to S" over all those S.

Under the Constraints of this problem, it can be proved that, for any string S of length N consisting of 0 and 1, there is at least one good path with respect to S.

Constraints

• 2 \leq N \leq 17
• N-1 \leq M \leq \frac{N(N-1)}{2}
• 1 \leq u_i, v_i \leq N
• The given graph is simple and connected.
• All values in input are integers.

Sample Input 1

3 2
1 2
2 3

Sample Output 1

14

• For S = 000, the empty sequence () is the shortest good path with respect to S, whose length is 0.
• For S = 100, (1) is the shortest good path with respect to S, whose length is 1.
• For S = 010, (2) is the shortest good path with respect to S, whose length is 1.
• For S = 110, (1, 2) is the shortest good path with respect to S, whose length is 2.
• For S = 001, (3) is the shortest good path with respect to S, whose length is 1.
• For S = 101, (1, 2, 3, 2) is the shortest good path with respect to S, whose length is 4.
• For S = 011, (2, 3) is the shortest good path with respect to S, whose length is 2.
• For S = 111, (1, 2, 3) is the shortest good path with respect to S, whose length is 3.

Therefore, the sought answer is 0 + 1 + 1 + 2 + 1 + 4 + 2 + 3 = 14.

Sample Input 2

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

Sample Output 2

108