Problem Statement

Print an arithmetic sequence with first term A, last term B, and common difference D.

You are only given inputs for which such an arithmetic sequence exists.

Constraints

• 1 \leq A \leq B \leq 100
• 1 \leq D \leq 100
• There is an arithmetic sequence with first term A, last term B, and common difference D.
• All input values are integers.

Sample Input 1

3 9 2

Sample Output 1

3 5 7 9

The arithmetic sequence with first term 3, last term 9, and common difference 2 is (3,5,7,9).

Sample Input 2

10 10 1

Sample Output 2

10

The arithmetic sequence with first term 10, last term 10, and common difference 1 is (10).

Problem Statement

You are given N strings W_1,W_2,\dots,W_N consisting of lowercase English letters.
If one or more of these strings equal and, not, that, the, or you, then print Yes; otherwise, print No.

Constraints

• N is an integer between 1 and 100, inclusive.
• 1 \le |W_i| \le 50 (|W_i| is the length of W_i.)
• W_i consists of lowercase English letters.

Sample Input 1

10
in that case you should print yes and not no

Sample Output 1

Yes

We have, for instance, W_4= you, so you should print Yes.

Sample Input 2

10
in diesem fall sollten sie no und nicht yes ausgeben

Sample Output 2

No

None of the strings W_i equals any of and, not, that, the, and you.

Problem Statement

You are given two positive integers N and M.

Let X = \displaystyle\sum_{i = 0}^{M} N^i. If X \leq 10^9, print the value of X. If X > 10^9, print inf.

Constraints

• 1 \leq N \leq 10^9
• 1 \leq M \leq 100
• All input values are integers.

Sample Input 1

7 3

Sample Output 1

400

X = 1 + 7 + 49 + 343 = 400. Since 400 \leq 10^9, print 400.

Sample Input 2

1000000 2

Sample Output 2

inf

X = 1000001000001 > 10^9, so print inf.

Sample Input 3

999999999 1

Sample Output 3

1000000000

Sample Input 4

998244353 99

Sample Output 4

inf

Problem Statement

There are N customers numbered 1 to N, and M canned juices numbered 1 to M.

Customer i (1 \leq i \leq N) has a wish list of length L_i. The j-th item (1 \leq j \leq L_i) from the top of customer i's wish list is canned juice X_{i,j}. For any customer i, the numbers X_{i, 1}, \dots, X_{i, L_i} on customer i's wish list are distinct.

Customers 1, \dots, N, in this order, will now choose their beverages, following the procedure below.

• If the customer's wish list contains a canned juice that has not yet been chosen by anyone at that point, they choose the canned juice whose number appears earliest in their wish list. Otherwise, they choose water.

Determine which beverage each customer gets.

Constraints

• 1 \leq N \leq 100
• 1 \leq M \leq 100
• 1 \leq L_i \leq M (1 \leq i \leq N)
• 1 \leq X_{i,j} \leq M (1 \leq i \leq N, 1 \leq j \leq L_i)
• X_{i, 1}, \dots, X_{i, L_i} are distinct. (1 \leq i \leq N)
• All input values are integers.

Sample Input 1

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

Sample Output 1

3
2
0
5

Among the numbers on customer 1's wish list, the canned juices not yet chosen by anyone are 3, 1, 2. The one appearing earliest in the list is 3, so customer 1 chooses canned juice 3.

Among the numbers on customer 2's wish list, the canned juices not yet chosen by anyone are 2, 1. The one appearing earliest in the list is 2, so customer 2 chooses canned juice 2.

For the numbers on customer 3's wish list, all corresponding canned juices have already been chosen by someone at that point. Thus, customer 3 chooses water.

Among the numbers on customer 4's wish list, the canned juices not yet chosen by anyone are 5, 1. The one appearing earliest in the list is 5, so customer 4 chooses canned juice 5.

Sample Input 2

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

Sample Output 2

3
5
1
4
2
0

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

Takahashi has a red jewel of level N. (He has no other jewels.)
Takahashi can do the following operations any number of times.

• Convert a red jewel of level n (n is at least 2) into "a red jewel of level (n-1) and X blue jewels of level n".
• Convert a blue jewel of level n (n is at least 2) into "a red jewel of level (n-1) and Y blue jewels of level (n-1)".

Takahashi wants as many blue jewels of level 1 as possible. At most, how many blue jewels of level 1 can he obtain by the operations?

Constraints

• 1 \leq N \leq 10
• 1 \leq X \leq 5
• 1 \leq Y \leq 5
• All values in input are integers.

Sample Input 1

2 3 4

Sample Output 1

12

Takahashi can obtain 12 blue jewels of level 1 by the following conversions.

• First, he converts a red jewel of level 2 into a red jewel of level 1 and 3 blue jewels of level 2.
  • After this operation, Takahashi has 1 red jewel of level 1 and 3 blue jewels of level 2.
• Next, he repeats the following conversion 3 times: converting a blue jewel of level 2 into a red jewel of level 1 and 4 blue jewels of level 1.
  • After these operations, Takahashi has 4 red jewels of level 1 and 12 blue jewels of level 1.
• He cannot perform a conversion anymore.

He cannot obtain more than 12 blue jewels of level 1, so the answer is 12.

Sample Input 2

1 5 5

Sample Output 2

0

Takahashi may not be able to obtain a blue jewel of level 1.

Sample Input 3

10 5 5

Sample Output 3

3942349900

Note that the answer may not fit into a 32-bit integer type.

Problem Statement

There is a simple undirected graph G with N vertices and M edges. The vertices are numbered 1, 2, \ldots, N, and the i-th edge is an undirected edge connecting vertices A_i and B_i.

You can repeat the following two operations in any order and any number of times:

• Add one undirected edge to G
• Remove one undirected edge from G

Find the minimum number of operations to make G a simple undirected graph where all vertices have degree 2.

Constraints

• 3 \leq N \leq 8
• 0 \leq M \leq \frac{N(N-1)}{2}
• The graph G given in the input is a simple undirected graph.
• All input values are integers.

Sample Input 1

5 4
1 2
1 5
2 4
4 5

Sample Output 1

3

For example, the following three operations make G a simple undirected graph where all vertices have degree 2.

• Add an edge connecting vertices 2 and 3.
• Remove the edge connecting vertices 2 and 4.
• Add an edge connecting vertices 3 and 4.

Sample Input 2

3 0

Sample Output 2

3

Sample Input 3

6 8
1 4
1 5
2 3
2 6
3 4
3 6
4 5
4 6

Sample Output 3

2

Sample Input 4

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

Sample Output 4

13

Problem Statement

Consider the following procedure of, given a string X consisting of lowercase English alphabets, obtaining a new string:

• Split the string X off at the positions where two different characters are adjacent to each other.
• For each string Y that has been split off, replace Y with a string consisting of the character which Y consists of, followed by the length of Y.
• Concatenate the replaced strings without changing the order.

For example, aaabbcccc is divided into aaa,bb,cccc, which are replaced by a3,b2,c4, respectively, which in turn are concatenated without changing the order, resulting in a3b2c4.If the given string is aaaaaaaaaa , the new string is a10 .

Find the number, modulo P, of strings S of lengths N consisting of lowercase English alphabets, such that the length of T is smaller than that of S, where T is the string obtained by the procedure above against the string S.

Constraints

• 1 \le N \le 3000
• 10^8 \le P \le 10^9
• N and P are integers.
• P is a prime.

Sample Input 1

3 998244353

Sample Output 1

26

Those strings of which the 1-st, 2-nd, and 3-rd characters are all the same satisfy the condition.

For example, aaa becomes a3, which satisfies the condition, while abc becomes a1b1c1, which does not.

Sample Input 2

2 998244353

Sample Output 2

0

Note that if a string is transformed into another string of the same length, such as aa that becomes a2, it does not satisfy the condition.

Sample Input 3

5 998244353

Sample Output 3

2626

Strings like aaabb and aaaaa satisfy the condition.

Sample Input 4

3000 924844033

Sample Output 4

607425699

Find the number of strings satisfying the condition modulo P.

Problem Statement

There is a race through checkpoints 1,2,\dots,N in this order on a coordinate plane.
The coordinates of checkpoint i are (X_i,Y_i), and all checkpoints have different coordinates.

Checkpoints other than checkpoints 1 and N can be skipped.
However, let C be the number of checkpoints skipped, and the following penalty will be imposed:

• \displaystyle 2^{C−1} if C>0, and
• 0 if C=0.

Let s be the total distance traveled (Euclidean distance) from checkpoint 1 to checkpoint N plus the penalty.
Find the minimum achievable value as s.

Constraints

• All input values are integers.
• 2 \le N \le 10^4
• 0 \le X_i,Y_i \le 10^4
• (X_i,Y_i) \neq (X_j,Y_j) if i \neq j.

Sample Input 1

6
0 0
1 1
2 0
0 1
1 0
2 1

Sample Output 1

5.82842712474619009753

Consider passing through checkpoints 1,2,5,6 and skip checkpoints 3,4.

• Move from checkpoint 1 to 2. The distance between them is \sqrt{2}.
• Move from checkpoint 2 to 5. The distance between them is 1.
• Move from checkpoint 5 to 6. The distance between them is \sqrt{2}.
• Two checkpoints are skipped, so the penalty of 2 is imposed.

In this way, you can achieve s = 3 + 2\sqrt{2} \approx 5.828427.
You cannot make s smaller than this value.

Sample Input 2

10
1 8
3 7
9 4
4 9
6 1
7 5
0 0
1 3
6 8
6 4

Sample Output 2

24.63441361516795872523

Sample Input 3

10
34 24
47 60
30 31
12 97
87 93
64 46
82 50
14 7
17 24
3 78

Sample Output 3

110.61238353245736230207