Problem Statement

The magnitude of an earthquake is a logarithmic scale of the energy released by the earthquake. It is known that each time the magnitude increases by 1, the amount of energy gets multiplied by approximately 32.
Here, we assume that the amount of energy gets multiplied by exactly 32 each time the magnitude increases by 1. In this case, how many times is the amount of energy of a magnitude A earthquake as much as that of a magnitude B earthquake?

Constraints

• 3\leq B\leq A\leq 9
• A and B are integers.

Sample Input 1

6 4

Sample Output 1

1024

6 is 2 greater than 4, so a magnitude 6 earthquake has 32\times 32=1024 times as much energy as a magnitude 4 earthquake has.

Sample Input 2

5 5

Sample Output 2

1

Earthquakes with the same magnitude have the same amount of energy.

Problem Statement

Naohiro has a monster. The monster's current health is H.
He also has N kinds of potions, numbered from 1 to N in ascending order of effectiveness.
If you give the monster potion n, its health will increase by P_n. Here, P_1 \lt P_2 \lt \dots \lt P_N.

He wants to increase the monster's health to X or above by giving it one of the potions.
Print the number of the least effective potion that can achieve the purpose. (The constraints guarantee that such a potion exists.)

Constraints

• 2 \leq N \leq 100
• 1 \leq H \lt X \leq 999
• 1 \leq P_1 \lt P_2 \lt \dots \lt P_N = 999
• All input values are integers.

Sample Input 1

3 100 200
50 200 999

Sample Output 1

2

Below is the change in the monster's health when one of the potions is given to the monster.

• If potion 1 is given, the monster's health becomes 100 + 50 = 150.
• If potion 2 is given, the monster's health becomes 100 + 200 = 300.
• If potion 3 is given, the monster's health becomes 100 + 999 = 1099.

The potions that increase the monster's health to at least X = 200 are potions 2 and 3. The answer is the least effective of them, which is potion 2.

Sample Input 2

2 10 21
10 999

Sample Output 2

2

Sample Input 3

10 500 999
38 420 490 585 613 614 760 926 945 999

Sample Output 3

4

Problem Statement

N students took an exam. The students are labeled as Student 1, Student 2, \dots, Student N, and Student i scored a_i points.

A student who scored less than P points are considered to have failed the exam and cannot earn the credit. Find the number of students who failed the exam.

Constraints

• 1 \leq N \leq 10^5
• 1 \leq P \leq 100
• 0 \leq a_i \leq 100 (1 \leq i \leq N)
• All values in input are integers.

Sample Input 1

4 50
80 60 40 0

Sample Output 1

2

Students 1 and 2, who scored 80 and 60 points, respectively, succeeded in scoring at least 50 points to earn the credit.
On the other hand, Students 3 and 4, who scored 40 and 0 points, respectively, fell below 50 points and failed the exam. Thus, the answer is 2.

Sample Input 2

3 90
89 89 89

Sample Output 2

3

Sample Input 3

2 22
6 37

Sample Output 3

1

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

You are given a string S of length N consisting of lowercase English letters, ,, and ". It is guaranteed that S contains an even number of ".

Let 2K be the number of " in S. For each i=1,2,\ldots,K, the characters from the (2i-1)-th " through the (2i)-th " are said to be enclosed.

Your task is to replace each , in S that is not an enclosed character with . and print the resulting string.

Constraints

• N is an integer between 1 and 2\times 10^5, inclusive.
• S is a string of length N consisting of lowercase English letters, ,, and ".
• S contains an even number of ".

Sample Input 1

8
"a,b"c,d

Sample Output 1

"a,b"c.d

In S, "a,b" are enclosed characters, and c,d are not.

The , in S that is not an enclosed character is the seventh character from the left in S, so replace that character with . to get the answer.

Sample Input 2

5
,,,,,

Sample Output 2

.....

Sample Input 3

20
a,"t,"c,"o,"d,"e,"r,

Sample Output 3

a."t,"c."o,"d."e,"r.

Problem Statement

We have a sequence of N numbers: A = (a_1, a_2, \dots, a_N).
Process the Q queries explained below.

• Query i: You are given a pair of integers (x_i, k_i). Let us look at the elements of A one by one from the beginning: a_1, a_2, \dots Which element will be the k_i-th occurrence of the number x_i?
  Print the index of that element, or -1 if there is no such element.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq Q \leq 2 \times 10^5
• 0 \leq a_i \leq 10^9 (1 \leq i \leq N)
• 0 \leq x_i \leq 10^9 (1 \leq i \leq Q)
• 1 \leq k_i \leq N (1 \leq i \leq Q)
• All values in input are integers.

Sample Input 1

6 8
1 1 2 3 1 2
1 1
1 2
1 3
1 4
2 1
2 2
2 3
4 1

Sample Output 1

1
2
5
-1
3
6
-1
-1

1 occurs in A at a_1, a_2, a_5. Thus, the answers to Query 1 through 4 are 1, 2, 5, -1 in this order.

Sample Input 2

3 2
0 1000000000 999999999
1000000000 1
123456789 1

Sample Output 2

2
-1

Problem Statement

Takahashi found a puzzle along some road.
It is composed of an undirected graph with nine vertices and M edges, and eight pieces.

The nine vertices of the graph are called Vertex 1, Vertex 2, \ldots, Vertex 9. For each i = 1, 2, \ldots, M, the i-th edge connects Vertex u_i and Vertex v_i.
The eight pieces are called Piece 1, Piece 2, \ldots, Piece 8. For each j = 1, 2, \ldots, 8, Piece j is on Vertex p_j.
Here, it is guaranteed that all pieces are on distinct vertices. Note that there is exactly one empty vertex without a piece.

Takahashi can do the following operation on the puzzle any number of times (possibly zero).

Choose a piece on a vertex adjacent to the empty vertex, and move it to the empty vertex.

By repeating this operation, he aims to complete the puzzle. The puzzle is considered complete when the following holds.

• For each j = 1, 2, \ldots, 8, Piece j is on Vertex j.

Determine whether it is possible for Takahashi to complete the puzzle. If it is possible, find the minimum number of operations needed to do so.

Constraints

• 0 \leq M \leq 36
• 1 \leq u_i, v_i \leq 9
• The given graph has no multi-edges or self-loops.
• 1 \leq p_j \leq 9
• j \neq j' \Rightarrow p_j \neq p_{j'}
• All values in input are integers.

Sample Input 1

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

Sample Output 1

5

The following procedure completes the puzzle in five operations.

1. Move Piece 2 from Vertex 9 to Vertex 1.
2. Move Piece 3 from Vertex 2 to Vertex 9.
3. Move Piece 2 from Vertex 1 to Vertex 2.
4. Move Piece 1 from Vertex 3 to Vertex 1.
5. Move Piece 3 from Vertex 9 to Vertex 3.

On the other hand, it is impossible to complete the puzzle in less than five operations. Thus, we should print 5.
Note that the given graph may not be connected.

Sample Input 2

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

Sample Output 2

0

The puzzle is already complete from the beginning. Thus, the minimum number of operations needed to complete the puzzle is 0.

Sample Input 3

12
8 5
9 6
4 5
4 1
2 5
8 9
2 1
3 6
8 7
6 5
7 4
2 3
1 2 3 4 5 6 8 7

Sample Output 3

-1

No sequence of operations can complete the puzzle, so we should print -1.

Sample Input 4

12
6 5
5 4
4 1
4 7
8 5
2 1
2 5
6 9
3 6
9 8
8 7
3 2
2 3 4 6 1 9 7 8

Sample Output 4

16

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. The degree of each vertex is at most 10.
Let K be the number of simple paths (paths without repeated vertices) starting from vertex 1. Print \min(K, 10^6).

Constraints

• 1 \leq N \leq 2 \times 10^5
• 0 \leq M \leq \min \left(2 \times 10^5, \frac{N(N-1)}{2}\right)
• 1 \leq u_i, v_i \leq N
• The given graph is simple.
• The degree of each vertex in the given graph is at most 10.
• All values in the input are integers.

Sample Input 1

4 2
1 2
2 3

Sample Output 1

3

We have the following three paths that count. (Note that a path of length 0 also counts.)

• Vertex 1;
• vertex 1, vertex 2;
• vertex 1, vertex 2, vertex 3.

Sample Input 2

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

Sample Output 2

16

Sample Input 3

8 21
2 6
1 3
5 6
3 8
3 6
4 7
4 6
3 4
1 5
2 4
1 2
2 7
1 4
3 5
2 5
2 3
4 5
3 7
6 7
5 7
2 8

Sample Output 3

2023

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