Problem Statement

Two characters x and y are called similar characters if and only if one of the following conditions is satisfied:

• x and y are the same character.
• One of x and y is 1 and the other is l.
• One of x and y is 0 and the other is o.

Two strings S and T, each of length N, are called similar strings if and only if:

• for all i\ (1\leq i\leq N), the i-th character of S and the i-th character of T are similar characters.

Given two length-N strings S and T consisting of lowercase English letters and digits, determine if S and T are similar strings.

Constraints

• N is an integer between 1 and 100.
• Each of S and T is a string of length N consisting of lowercase English letters and digits.

Sample Input 1

3
l0w
1ow

Sample Output 1

Yes

The 1-st character of S is l, and the 1-st character of T is 1. These are similar characters.

The 2-nd character of S is 0, and the 2-nd character of T is o. These are similar characters.

The 3-rd character of S is w, and the 3-rd character of T is w. These are similar characters.

Thus, S and T are similar strings.

Sample Input 2

3
abc
arc

Sample Output 2

No

The 2-nd character of S is b, and the 2-nd character of T is r. These are not similar characters.

Thus, S and T are not similar strings.

Sample Input 3

4
nok0
n0ko

Sample Output 3

Yes

Problem Statement

Given integers a, b, and c, determine if b is the median of these integers.

Constraints

• 1 \leq a, b, c \leq 100
• All values in input are integers.

Sample Input 1

5 3 2

Sample Output 1

Yes

The given integers are 2, 3, 5 when sorted in ascending order, of which b is the median.

Sample Input 2

2 5 3

Sample Output 2

No

b is not the median of the given integers.

Sample Input 3

100 100 100

Sample Output 3

Yes

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

N players with ID numbers 1, 2, \ldots, N are playing a card game.
Each player plays one card.

Each card has two parameters: color and rank, both of which are represented by positive integers.
For i = 1, 2, \ldots, N, the card played by player i has a color C_i and a rank R_i. All of R_1, R_2, \ldots, R_N are different.

Among the N players, one winner is decided as follows.

• If one or more cards with the color T are played, the player who has played the card with the greatest rank among those cards is the winner.
• If no card with the color T is played, the player who has played the card with the greatest rank among the cards with the color of the card played by player 1 is the winner. (Note that player 1 may win.)

Print the ID number of the winner.

Constraints

• 2 \leq N \leq 2 \times 10^5
• 1 \leq T \leq 10^9
• 1 \leq C_i \leq 10^9
• 1 \leq R_i \leq 10^9
• i \neq j \implies R_i \neq R_j
• All values in the input are integers.

Sample Input 1

4 2
1 2 1 2
6 3 4 5

Sample Output 1

4

Cards with the color 2 are played. Thus, the winner is player 4, who has played the card with the greatest rank, 5, among those cards.

Sample Input 2

4 2
1 3 1 4
6 3 4 5

Sample Output 2

1

No card with the color 2 is played. Thus, the winner is player 1, who has played the card with the greatest rank, 6, among the cards with the color of the card played by player 1 (color 1).

Sample Input 3

2 1000000000
1000000000 1
1 1000000000

Sample Output 3

1

Problem Statement

You are given an integer sequence A=(A_1,A_2,\dots,A_N). You can perform the following operation any number of times (possibly zero).

• Choose integers i and j with 1\leq i,j \leq N. Decrease A_i by one and increase A_j by one.

Find the minimum number of operations required to make the difference between the minimum and maximum values of A at most one.

Constraints

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

Sample Input 1

4
4 7 3 7

Sample Output 1

3

By the following three operations, the difference between the minimum and maximum values of A becomes at most one.

• Choose i=2 and j=3 to make A=(4,6,4,7).
• Choose i=4 and j=1 to make A=(5,6,4,6).
• Choose i=4 and j=3 to make A=(5,6,5,5).

You cannot make the difference between maximum and minimum values of A at most one by less than three operations, so the answer is 3.

Sample Input 2

1
313

Sample Output 2

0

Sample Input 3

10
999999997 999999999 4 3 2 4 999999990 8 999999991 999999993

Sample Output 3

2499999974

Problem Statement

A single-player card game is popular in AtCoder Inc. Each card in the game has a lowercase English letter or the symbol @ written on it. There is plenty number of cards for each kind. The game goes as follows.

1. Arrange the same number of cards in two rows.
2. Replace each card with @ with one of the following cards: a, t, c, o, d, e, r.
3. If the two rows of cards coincide, you win. Otherwise, you lose.

To win this game, you will do the following cheat.

• Freely rearrange the cards within a row whenever you want after step 1.

You are given two strings S and T, representing the two rows you have after step 1. Determine whether it is possible to win with cheating allowed.

Constraints

• S and T consist of lowercase English letters and @.
• The lengths of S and T are equal and between 1 and 2\times 10^5, inclusive.

Sample Input 1

ch@ku@ai
choku@@i

Sample Output 1

Yes

You can replace the @s so that both rows become chokudai.

Sample Input 2

ch@kud@i
akidu@ho

Sample Output 2

Yes

You can cheat and replace the @s so that both rows become chokudai.

Sample Input 3

aoki
@ok@

Sample Output 3

No

You cannot win even with cheating.

Sample Input 4

aa
bb

Sample Output 4

No

Problem Statement

We have a positive integer a. Additionally, there is a blackboard with a number written in base 10.
Let x be the number on the blackboard. Takahashi can do the operations below to change this number.

• Erase x and write x multiplied by a, in base 10.
• See x as a string and move the rightmost digit to the beginning.
  This operation can only be done when x \geq 10 and x is not divisible by 10.

For example, when a = 2, x = 123, Takahashi can do one of the following.

• Erase x and write x \times a = 123 \times 2 = 246.
• See x as a string and move the rightmost digit 3 of 123 to the beginning, changing the number from 123 to 312.

The number on the blackboard is initially 1. What is the minimum number of operations needed to change the number on the blackboard to N? If there is no way to change the number to N, print -1.

Constraints

• 2 \leq a \lt 10^6
• 2 \leq N \lt 10^6
• All values in input are integers.

Sample Input 1

3 72

Sample Output 1

4

We can change the number on the blackboard from 1 to 72 in four operations, as follows.

• Do the operation of the first type: 1 \to 3.
• Do the operation of the first type: 3 \to 9.
• Do the operation of the first type: 9 \to 27.
• Do the operation of the second type: 27 \to 72.

It is impossible to reach 72 in three or fewer operations, so the answer is 4.

Sample Input 2

2 5

Sample Output 2

-1

It is impossible to change the number on the blackboard to 5.

Sample Input 3

2 611

Sample Output 3

12

There is a way to change the number on the blackboard to 611 in 12 operations: 1 \to 2 \to 4 \to 8 \to 16 \to 32 \to 64 \to 46 \to 92 \to 29 \to 58 \to 116 \to 611, which is the minimum possible.

Sample Input 4

2 767090

Sample Output 4

111

Problem Statement

In the nation of AtCoder, there are N cities numbered 1 to N and N-1 roads numbered 1 to N-1.

Road i connects cities A_i and B_i bidirectionally, and its length is C_i. Any pair of cities can be reached from each other by traveling through some roads.

Find the minimum travel distance required to start from a city and visit all cities at least once using the roads.

Constraints

• 2 \leq N \leq 2\times 10^5
• 1 \leq A_i, B_i \leq N
• 1 \leq C_i \leq 10^9
• All input values are integers.
• Any pair of cities can be reached from each other by traveling through some roads.

Sample Input 1

4
1 2 2
1 3 3
1 4 4

Sample Output 1

11

If you travel as 4 \to 1 \to 2 \to 1 \to 3, the total travel distance is 11, which is the minimum.

Note that you do not need to return to the starting city.

Sample Input 2

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

Sample Output 2

9000000000

Beware overflow.

Problem Statement

N players, player 1, player 2, ..., player N, participate in a game tournament. Just before the tournament starts, each player forms a one-person team, so there are N teams in total.

The tournament has a total of N-1 matches. In each match, two different teams are chosen. One team goes first, and the other goes second. Each match will result in exactly one team winning. Specifically, for each i = 1, 2, \ldots, N-1, the i-th match proceeds as follows.

• The team with player p_i goes first, and the team with player q_i goes second.
• Let a and b be the numbers of players in the first and second teams, respectively. The first team wins with probability \frac{a}{a+b}, and the second team wins with probability \frac{b}{a+b}.
• Then, the two teams are combined into a single team.

The result of each match is independent of those of the others.

For each of the N players, print the expected number of times the team with that player wins throughout the tournament, modulo 998244353.

Constraints

• 2 \leq N \leq 2 \times 10^5
• 1 \leq p_i, q_i \leq N
• Just before the i-th match, player p_i and player q_i belong to different teams.
• All input values are integers.

Sample Input 1

5
1 2
4 3
5 3
1 4

Sample Output 1

698771048 698771048 964969543 964969543 133099248

We call a team formed by player x_1, player x_2, \ldots, player x_k as team \lbrace x_1, x_2, \ldots, x_k \rbrace.

• The first match is played by team \lbrace 1 \rbrace, with player 1, and team \lbrace 2 \rbrace, with player 2. Team \lbrace 1 \rbrace wins with probability \frac{1}{2}, and team \lbrace 2 \rbrace wins with probability \frac{1}{2}. Then, the two teams are combined into a single team \lbrace 1, 2 \rbrace.
• The second match is played by team \lbrace 4 \rbrace, with player 4, and team \lbrace 3 \rbrace, with player 3. Team \lbrace 4 \rbrace wins with probability \frac{1}{2}, and team \lbrace 3 \rbrace wins with probability \frac{1}{2}. Then, the two teams are combined into a single team \lbrace 3, 4 \rbrace.
• The third match is played by team \lbrace 5 \rbrace, with player 5, and team \lbrace 3, 4 \rbrace, with player 3. Team \lbrace 5 \rbrace wins with probability \frac{1}{3}, and team \lbrace 3, 4 \rbrace wins with probability \frac{2}{3}. Then, the two teams are combined into a single team \lbrace 3, 4, 5 \rbrace.
• The fourth match is played by team \lbrace 1, 2 \rbrace, with player 1, and team \lbrace 3, 4, 5 \rbrace, with player 4. Team \lbrace 1, 2 \rbrace wins with probability \frac{2}{5}, and team \lbrace 3, 4, 5 \rbrace wins with probability \frac{3}{5}. Then, the two teams are combined into a single team \lbrace 1, 2, 3, 4, 5 \rbrace.

The expected numbers of times the teams with players 1, 2, 3, 4, 5 win throughout the tournament, E_1, E_2, E_3, E_4, E_5, are \frac{9}{10}, \frac{9}{10}, \frac{53}{30}, \frac{53}{30}, \frac{14}{15}, respectively.

Sample Input 2

15
9 2
8 10
13 6
12 11
7 10
4 10
14 2
5 4
1 15
15 2
6 9
8 11
6 3
2 8

Sample Output 2

43970290 310168785 806914186 501498951 950708909 272140427 335124893 168750835 310168785 168750835 280459129 280459129 272140427 476542843 43970290