Problem Statement

You are given an integer K.

Print a string that is a concatenation of the first K uppercase English letters in ascending order, starting from A.

Constraints

• K is an integer between 1 and 26, inclusive.

Sample Input 1

3

Sample Output 1

ABC

The uppercase English letters in ascending order are A, B, C, ...

By concatenating the first three uppercase English letters, we get ABC.

Sample Input 2

1

Sample Output 2

A

Problem Statement

A fruit store sells apples.
You may perform the following operations as many times as you want in any order:

• Buy one apple for X yen (the currency in Japan).
• Buy three apples for Y yen.

How much yen do you need to pay to obtain exactly N apples?

Constraints

• 1 \leq X \leq Y \leq 100
• 1 \leq N \leq 100
• All values in input are integers.

Sample Input 1

10 25 10

Sample Output 1

85

Buy three apples for 25 yen three times and one apple for 10 yen, and you will obtain exactly 10 apples for a total of 85 yen.
You cannot obtain exactly 10 apples for a lower cost, so the answer is 85 yen.

Sample Input 2

10 40 10

Sample Output 2

100

It is optimal to buy an apple for 10 yen 10 times.

Sample Input 3

100 100 2

Sample Output 3

200

The only way to obtain exactly 2 apples is to buy an apple for 100 yen twice.

Sample Input 4

100 100 100

Sample Output 4

3400

Problem Statement

You are given integers A and B, in base K.
Print A \times B in decimal.

Notes

For base-K representation, see Wikipedia's article on Positional notation.

Constraints

• 2 \leq K \leq 10
• 1 \leq A,B \leq 10^5
• A and B are in base-K representation.

Sample Input 1

2
1011 10100

Sample Output 1

220

1011 in base 2 corresponds to 11 in base 10.
10100 in base 2 corresponds to 20 in base 10.
We have 11 \times 20 = 220, so print 220.

Sample Input 2

7
123 456

Sample Output 2

15642

123 in base 7 corresponds to 66 in base 10.
456 in base 7 corresponds to 237 in base 10.
We have 66 \times 237 = 15642, so print 15642.

Problem Statement

There are N people, called Person 1, Person 2, \ldots, Person N.

The parent of Person i (2 \le i \le N) is Person P_i. Here, it is guaranteed that P_i < i.

How many generations away from Person N is Person 1?

Constraints

• 2 \le N \le 50
• 1 \le P_i < i(2 \le i \le N)
• All values in input are integers.

Sample Input 1

3
1 2

Sample Output 1

2

Person 2 is a parent of Person 3, and thus is one generation away from Person 3.

Person 1 is a parent of Person 2, and thus is two generations away from Person 3.

Therefore, the answer is 2.

Sample Input 2

10
1 2 3 4 5 6 7 8 9

Sample Output 2

9

Problem Statement

Given is a string S consisting of lowercase English letters. Determine whether adding some number of a's (possibly zero) at the beginning of S can make it a palindrome.

Here, a string of length N, A=A_1A_2\ldots A_N, is said to be a palindrome when A_i=A_{N+1-i} for every 1\leq i\leq N.

Constraints

• 1 \leq \lvert S \rvert \leq 10^6
• S consists of lowercase English letters.

Sample Input 1

kasaka

Sample Output 1

Yes

By adding one a at the beginning of kasaka, we have akasaka, which is a palindrome, so Yes should be printed.

Sample Input 2

atcoder

Sample Output 2

No

Adding any number of a's at the beginning of atcoder does not make it a palindrome.

Sample Input 3

php

Sample Output 3

Yes

php itself is a palindrome. Adding zero a's at the beginning of S is allowed, so Yes should be printed.

Problem Statement

We have N fuses connected in series. The i-th fuse from the left has a length of A_i centimeters and burns at a constant speed of B_i centimeters per second.

Consider igniting this object from left and right ends simultaneously. Find the distance between the position where the two flames will meet and the left end of the object.

Constraints

• 1 \leq N \leq 10^5
• 1 \leq A_i,B_i \leq 1000
• All values in input are integers.

Sample Input 1

3
1 1
2 1
3 1

Sample Output 1

3.000000000000000

The two flames will meet at 3 centimeters from the left end of the object.

Sample Input 2

3
1 3
2 2
3 1

Sample Output 2

3.833333333333333

Sample Input 3

5
3 9
1 2
4 6
1 5
5 3

Sample Output 3

8.916666666666668

Problem Statement

We have a video game consisting of N stages. The i-th stage (1 \leq i \leq N) is composed of a movie lasting A_i minutes and gameplay lasting B_i minutes.

To clear the i-th stage for the first time, one must watch the movie and do the gameplay for that stage. For the second and subsequent times, one may skip the movie and do just the gameplay.

In the beginning, only the 1-st stage is unlocked, and clearing the i-th stage (1 \leq i \leq N - 1) unlocks the (i+1)-th stage.

Find the shortest time needed to clear a stage X times in total. Here, if the same stage is cleared multiple times, all of them count.

Constraints

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

Sample Input 1

3 4
3 4
2 3
4 2

Sample Output 1

18

Here is one way to clear a stage 4 times in 18 minutes:

• Clear Stage 1. It takes A_1 + B_1 = 7 minutes.
• Clear Stage 2. It takes A_2 + B_2 = 5 minutes.
• Clear Stage 2 again. It takes B_2= 3 minutes.
• Clear Stage 2 again. It takes B_2= 3 minutes.

It is impossible to clear a stage 4 times within 17 minutes.

Sample Input 2

10 1000000000
3 3
1 6
4 7
1 8
5 7
9 9
2 4
6 4
5 1
3 1

Sample Output 2

1000000076

Problem Statement

There are N 7's drawn in the first quadrant of a plane.

The i-th 7 is a figure composed of a segment connecting (x_i-1,y_i) and (x_i,y_i), and a segment connecting (x_i,y_i-1) and (x_i,y_i).

You can choose zero or more from the N 7's and delete them.

What is the maximum possible number of 7's that are wholly visible from the origin after the optimal deletion?

Here, the i-th 7 is wholly visible from the origin if and only if:

• the interior (excluding borders) of the quadrilateral whose vertices are the origin, (x_i-1,y_i), (x_i,y_i), (x_i,y_i-1) does not intersect with the other 7's.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 1 \leq x_i,y_i \leq 10^9
• (x_i,y_i) \neq (x_j,y_j)\ (i \neq j)
• All values in input are integers.

Sample Input 1

3
1 1
2 1
1 2

Sample Output 1

2

If the first 7 is deleted, the other two 7's ― the second and third ones ― will be wholly visible from the origin, which is optimal.

If no 7's are deleted, only the first 7 is wholly visible from the origin.

Sample Input 2

10
414598724 87552841
252911401 309688555
623249116 421714323
605059493 227199170
410455266 373748111
861647548 916369023
527772558 682124751
356101507 249887028
292258775 110762985
850583108 796044319

Sample Output 2

10

It is best to keep all 7's.

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.