Problem Statement

There are N toys numbered from 1 to N, and N-1 boxes numbered from 1 to N-1. Toy i\ (1 \leq i \leq N) has a size of A_i, and box i\ (1 \leq i \leq N-1) has a size of B_i.

Takahashi wants to store all the toys in separate boxes, and he has decided to perform the following steps in order:

1. Choose an arbitrary positive integer x and purchase one box of size x.
2. Place each of the N toys into one of the N boxes (the N-1 existing boxes plus the newly purchased box). Here, each toy can only be placed in a box whose size is not less than the toy's size, and no box can contain two or more toys.

He wants to execute step 2 by purchasing a sufficiently large box in step 1, but larger boxes are more expensive, so he wants to purchase the smallest possible box.

Determine whether there exists a value of x such that he can execute step 2, and if it exists, find the minimum such x.

Constraints

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

Sample Input 1

4
5 2 3 7
6 2 8

Sample Output 1

3

Consider the case where x=3 (that is, he purchases a box of size 3 in step 1).

If the newly purchased box is called box 4, toys 1,\dots,4 have sizes of 5, 2, 3, and 7, respectively, and boxes 1,\dots,4 have sizes of 6, 2, 8, and 3, respectively. Thus, toy 1 can be placed in box 1, toy 2 in box 2, toy 3 in box 4, and toy 4 in box 3.

On the other hand, if x \leq 2, it is impossible to place all N toys into separate boxes. Therefore, the answer is 3.

Sample Input 2

4
3 7 2 5
8 1 6

Sample Output 2

-1

No matter what size of box is purchased in step 1, no toy can be placed in box 2, so it is impossible to execute step 2.

Sample Input 3

8
2 28 17 39 57 56 37 32
34 27 73 28 76 61 27

Sample Output 3

37

Problem Statement

For positive integers x and y, define f(x, y) as the remainder of (x + y) divided by 10^8.

You are given a sequence of positive integers A = (A_1, \ldots, A_N) of length N. Find the value of the following expression:

Constraints

• 2 \leq N \leq 3\times 10^5
• 1 \leq A_i < 10^8
• All input values are integers.

Sample Input 1

3
3 50000001 50000002

Sample Output 1

100000012

• f(A_1,A_2)=50000004
• f(A_1,A_3)=50000005
• f(A_2,A_3)=3

Thus, the answer is f(A_1,A_2) + f(A_1,A_3) + f(A_2,A_3) = 100000012.

Note that you are not asked to compute the remainder of the sum divided by 10^8.

Sample Input 2

5
1 3 99999999 99999994 1000000

Sample Output 2

303999988

Problem Statement

You are given a sequence of non-negative integers A=(a_1,a_2,\ldots,a_N).

Let S be the set of non-negative integers that can be the sum of K terms in A (with distinct indices).

Find the greatest multiple of D in S. If there is no multiple of D in S, print -1 instead.

Constraints

• 1 \leq K \leq N \leq 100
• 1 \leq D \leq 100
• 0 \leq a_i \leq 10^9
• All values in the input are integers.

Sample Input 1

4 2 2
1 2 3 4

Sample Output 1

6

Here are all the ways to choose two terms in A.

• Choose a_1 and a_2, whose sum is 1+2=3.
• Choose a_1 and a_3, whose sum is 1+3=4.
• Choose a_1 and a_4, whose sum is 1+4=5.
• Choose a_2 and a_3, whose sum is 2+3=5.
• Choose a_2 and a_4, whose sum is 2+4=6.
• Choose a_3 and a_4, whose sum is 3+4=7.

Thus, we have S=\{3,4,5,6,7\}. The greatest multiple of 2 in S is 6, so you should print 6.

Sample Input 2

3 1 2
1 3 5

Sample Output 2

-1

In this example, we have S=\{1,3,5\}. Nothing in S is a multiple of 2, so you should print -1.

Problem Statement

You are given strings A and B of equal length consisting of 0 and 1.

You can perform the following operation on A zero or more times.

• Move the first character of A to the end.

Find the minimum number of operations required to make A=B.
If it is impossible to make A=B no matter how you operate, print -1 instead.

You are given T test cases; find the answer for each of them.

Constraints

• 1 \le T \le 10000
• A and B are strings consisting of 0 and 1.
• 2 \le |A|=|B| \le 10^6
• For a single input, the sum of |A| does not exceed 10^6.

Sample Input 1

5
1010001
1000110
000
111
01010
01010
0101
0011
100001101110000001010110110001
101100011000011011100000010101

Sample Output 1

2
-1
0
-1
22

This input contains five test cases.

• For the first test case, A= 1010001 and B= 1000110.
  • By performing the operation on A twice, A becomes 1010001 \rightarrow 0100011 \rightarrow 1000110, which makes A=B.
• For the second test case, no matter how you perform the operation, you cannot change 000 to 111.
• For the third test case, A=B from the beginning.

Problem Statement

There are N balls arranged from left to right. The color of the i-th ball from the left is Color C_i, and an integer X_i is written on it.

Takahashi wants to rearrange the balls so that the integers written on the balls are non-decreasing from left to right. In other words, his objective is to reach a situation where, for every 1\leq i\leq N-1, the number written on the (i+1)-th ball from the left is greater than or equal to the number written on the i-th ball from the left.

For this, Takahashi can repeat the following operation any number of times (possibly zero):

Choose an integer i such that 1\leq i\leq N-1.
If the colors of the i-th and (i+1)-th balls from the left are different, pay a cost of 1. (No cost is incurred if the colors are the same).
Swap the i-th and (i+1)-th balls from the left.

Find the minimum total cost Takahashi needs to pay to achieve his objective.

Constraints

• 2 \leq N \leq 3\times 10^5
• 1\leq C_i\leq N
• 1\leq X_i\leq N
• All values in input are integers.

Sample Input 1

5
1 5 2 2 1
3 2 1 2 1

Sample Output 1

6

Let us represent a ball as (Color, Integer). The initial situation is (1,3), (5,2), (2,1), (2,2), (1,1). Here is a possible sequence of operations for Takahashi:

• Swap the 1-st ball (Color 1) and 2-nd ball (Color 5). Now the balls are arranged in the order (5,2), (1,3), (2,1), (2,2), (1,1).
• Swap the 2-nd ball (Color 1) and 3-rd ball (Color 2). Now the balls are arranged in the order (5,2), (2,1), (1,3), (2,2), (1,1).
• Swap the 3-rd ball (Color 1) and 4-th ball (Color 2). Now the balls are in the order (5,2), (2,1), (2,2), (1,3), (1,1).
• Swap the 4-th ball (Color 1) and 5-th ball (Color 1). Now the balls are in the order (5,2), (2,1), (2,2), (1,1), (1,3).
• Swap the 3-rd ball (Color 2) and 4-th ball (Color 1). Now the balls are in the order(5,2), (2,1), (1,1), (2,2), (1,3).
• Swap the 1-st ball (Color 5) and 2-nd ball (Color 2). Now the balls are in the order (2,1), (5,2), (1,1), (2,2), (1,3).
• Swap the 2-nd ball (Color 5) and 3-rd ball (Color 1). Now the balls are in the order (2,1), (1,1), (5,2), (2,2), (1,3).

After the last operation, the numbers written on the balls are 1,1,2,2,3 from left to right, which achieves Takahashi's objective.

The 1-st, 2-nd, 3-rd, 5-th, 6-th, and 7-th operations incur a cost of 1 each, for a total of 6, which is the minimum. Note that the 4-th operation does not incur a cost since the balls are both in Color 1.

Sample Input 2

3
1 1 1
3 2 1

Sample Output 2

0

All balls are in the same color, so no cost is incurred in swapping balls.

Sample Input 3

3
3 1 2
1 1 2

Sample Output 3

0

Takahashi's objective is already achieved without any operation.