Problem Statement

Assuming that the horizon seen from a place x meters above the ground is \sqrt{x(12800000+x)} meters away, find how many meters away the horizon seen from a place H meters above the ground is.

Constraints

• 1 \leq H \leq 10^5
• H is an integer.

Sample Input 1

333

Sample Output 1

65287.907678222

We have \sqrt{333(12800000+333)} = 65287.9076782\ldots. Outputs such as 65287.91 would also be accepted.

Sample Input 2

634

Sample Output 2

90086.635834623

We have \sqrt{634(12800000+634)} = 90086.6358346\ldots.

Problem Statement

You are given a string S of length exactly 9 consisting of digits. One but all digits from 0 to 9 appear exactly once in S.

Print the only digit missing in S.

Constraints

• S is a string of length 9 consisting of digits.
• All characters in S are distinct.

Sample Input 1

023456789

Sample Output 1

1

The string 023456789 only lacks 1. Thus, 1 should be printed.

Sample Input 2

459230781

Sample Output 2

6

The string 459230781 only lacks 6. Thus, 6 should be printed.

Note that the digits in the string may not appear in increasing order.

Problem Statement

There are N players numbered 1 to N, who have played a round-robin tournament. For every match in this tournament, one player won and the other lost.

The results of the matches are given as N strings S_1,S_2,\ldots,S_N of length N each, in the following format:

• If i\neq j, the j-th character of S_i is o or x. o means that player i won against player j, and x means that player i lost to player j.

• If i=j, the j-th character of S_i is -.

The player with more wins ranks higher. If two players have the same number of wins, the player with the smaller player number ranks higher. Report the player numbers of the N players in descending order of rank.

Constraints

• 2\leq N\leq 100
• N is an integer.
• S_i is a string of length N consisting of o, x, and -.
• S_1,\ldots,S_N conform to the format described in the problem statement.

Sample Input 1

3
-xx
o-x
oo-

Sample Output 1

3 2 1

Player 1 has 0 wins, player 2 has 1 win, and player 3 has 2 wins. Thus, the player numbers in descending order of rank are 3,2,1.

Sample Input 2

7
-oxoxox
x-xxxox
oo-xoox
xoo-ooo
ooxx-ox
xxxxx-x
oooxoo-

Sample Output 2

4 7 3 1 5 2 6

Both players 4 and 7 have 5 wins, but player 4 ranks higher because their player number is smaller.

Problem Statement

Takahashi is at the origin of a number line. He wants to reach a goal at coordinate X.

There is a wall at coordinate Y, which Takahashi cannot go beyond at first.
However, after picking up a hammer at coordinate Z, he can destroy that wall and pass through.

Determine whether Takahashi can reach the goal. If he can, find the minimum total distance he needs to travel to do so.

Constraints

• -1000 \leq X,Y,Z \leq 1000
• X, Y, and Z are distinct, and none of them is 0.
• All values in the input are integers.

Sample Input 1

10 -10 1

Sample Output 1

10

Takahashi can go straight to the goal.

Sample Input 2

20 10 -10

Sample Output 2

40

The goal is beyond the wall. He can get there by first picking up the hammer and then destroying the wall.

Sample Input 3

100 1 1000

Sample Output 3

-1

Problem Statement

There is a two-dimensional plane. For integers r and c between 1 and 9, there is a pawn at the coordinates (r,c) if the c-th character of S_{r} is #, and nothing if the c-th character of S_{r} is ..

Find the number of squares in this plane with pawns placed at all four vertices.

Constraints

• Each of S_1,\ldots,S_9 is a string of length 9 consisting of # and ..

Sample Input 1

##.......
##.......
.........
.......#.
.....#...
........#
......#..
.........
.........

Sample Output 1

2

The square with vertices (1,1), (1,2), (2,2), and (2,1) have pawns placed at all four vertices.

The square with vertices (4,8), (5,6), (7,7), and (6,9) also have pawns placed at all four vertices.

Thus, the answer is 2.

Sample Input 2

.#.......
#.#......
.#.......
.........
....#.#.#
.........
....#.#.#
........#
.........

Sample Output 2

3

Problem Statement

You are given a grid of H rows and W columns.
Let (i,j) denote the cell at row i (1 \leq i \leq H) from the top and column j (1 \leq j \leq W) from the left.
The state of the grid is represented by H strings S_1, S_2, \ldots, S_H, each of length W, as follows:

• If the j-th character of S_i is #, cell (i,j) is painted black.
• If the j-th character of S_i is ., cell (i,j) is painted white.
• If the j-th character of S_i is ?, cell (i,j) is not yet painted.

Takahashi wants to paint each not-yet-painted cell white or black so that all the black cells form a rectangle.
More precisely, he wants there to exist a quadruple of integers (a,b,c,d) (1 \leq a \leq b \leq H, 1 \leq c \leq d \leq W) such that:

For each cell (i,j) (1 \leq i \leq H, 1 \leq j \leq W), if a \leq i \leq b and c \leq j \leq d, the cell is black;
otherwise, the cell is white.

Determine whether this is possible.

Constraints

• 1 \leq H, W \leq 1000
• H and W are integers.
• Each S_i is a string of length W consisting of #, ., ?.
• There is at least one cell that is already painted black.

Sample Input 1

3 5
.#?#.
.?#?.
?...?

Sample Output 1

Yes

The grid is in the following state. ? indicates a cell that are not yet painted.

By painting cells (1,3), (2,2), and (2,4) black and cells (3,1) and (3,5) white, the black cells can form a rectangle as follows:

Therefore, print Yes.

Sample Input 2

3 3
?##
#.#
##?

Sample Output 2

No

To form a rectangle with all black cells, you would need to paint cell (2,2) black, but it is already painted white.
Therefore, it is impossible to make all black cells form a rectangle, so print No.

Sample Input 3

1 1
#

Sample Output 3

Yes

Problem Statement

2N people numbered 1, 2, \ldots, 2N attend a ball. They will group into N pairs and have a dance.

If Person i and Person j pair up, where i is smaller than j, the affinity of that pair is A_{i, j}.
If the N pairs have the affinity of B_1, B_2, \ldots, B_N, the total fun of the ball is the bitwise XOR of them: B_1 \oplus B_2 \oplus \cdots \oplus B_N.

Print the maximum possible total fun of the ball when the 2N people can freely group into N pairs.

Constraints

• 1 \leq N \leq 8
• 0 \leq A_{i, j} < 2^{30}
• All values in input are integers.

Sample Input 1

2
4 0 1
5 3
2

Sample Output 1

6

Let \lbrace i, j\rbrace denote a pair of Person i and Person j. There are three ways for the four people to group into two pairs, as follows.

• Group into \lbrace 1, 2\rbrace, \lbrace 3, 4\rbrace. The total fun of the ball here is A_{1, 2} \oplus A_{3, 4} = 4 \oplus 2 = 6.
• Group into \lbrace 1, 3\rbrace, \lbrace 2, 4\rbrace. The total fun of the ball here is A_{1, 3} \oplus A_{2, 4} = 0 \oplus 3 = 3.
• Group into \lbrace 1, 4\rbrace, \lbrace 2, 3\rbrace. The total fun of the ball here is A_{1, 4} \oplus A_{2, 3} = 1 \oplus 5 = 4.

Therefore, the maximum possible total fun of the ball is 6.

Sample Input 2

1
5

Sample Output 2

5

There will be just a pair of Person 1 and Person 2, where the total fun of the ball is 5.

Sample Input 3

5
900606388 317329110 665451442 1045743214 260775845 726039763 57365372 741277060 944347467
369646735 642395945 599952146 86221147 523579390 591944369 911198494 695097136
138172503 571268336 111747377 595746631 934427285 840101927 757856472
655483844 580613112 445614713 607825444 252585196 725229185
827291247 105489451 58628521 1032791417 152042357
919691140 703307785 100772330 370415195
666350287 691977663 987658020
1039679956 218233643
70938785

Sample Output 3

1073289207

Problem Statement

There is an undirected graph with N vertices and 0 edges. The vertices are numbered 1 to N.

You are given Q queries to process in order. Each query is of one of the following two types:

• Type 1: Given in the format 1 u v. Add an edge between vertices u and v.
• Type 2: Given in the format 2 v k. Print the k-th largest vertex number among the vertices connected to vertex v. If there are fewer than k vertices connected to v, print -1.

Constraints

• 1 \leq N, Q \leq 2 \times 10^5
• In a Type 1 query, 1 \leq u < v \leq N.
• In a Type 2 query, 1 \leq v \leq N, 1 \leq k \leq 10.
• All input values are integers.

Sample Input 1

4 10
1 1 2
2 1 1
2 1 2
2 1 3
1 1 3
1 2 3
1 3 4
2 1 1
2 1 3
2 1 5

Sample Output 1

2
1
-1
4
2
-1

• In the first query, an edge is added between vertices 1 and 2.
• In the second query, two vertices are connected to vertex 1: 1 and 2. Among them, the 1-st largest vertex number is 2, which should be printed.
• In the third query, two vertices are connected to vertex 1: 1 and 2. Among them, the 2-nd largest vertex number is 1, which should be printed.
• In the fourth query, two vertices are connected to vertex 1: 1 and 2, which is fewer than 3, so print -1.
• In the fifth query, an edge is added between vertices 1 and 3.
• In the sixth query, an edge is added between vertices 2 and 3.
• In the seventh query, an edge is added between vertices 3 and 4.
• In the eighth query, four vertices are connected to vertex 1: 1,2,3,4. Among them, the 1-st largest vertex number is 4, which should be printed.
• In the ninth query, four vertices are connected to vertex 1: 1,2,3,4. Among them, the 3-rd largest vertex number is 2, which should be printed.
• In the tenth query, four vertices are connected to vertex 1: 1,2,3,4, which is fewer than 5, so print -1.

Sample Input 2

6 20
1 3 4
1 3 5
2 1 1
2 3 1
1 1 5
2 6 9
2 1 3
2 6 1
1 4 6
2 2 1
2 6 2
2 4 7
1 1 4
2 6 2
2 3 4
1 2 5
2 4 1
1 1 6
2 3 3
2 1 3

Sample Output 2

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

Problem Statement

Given are M digits C_i.

Find the sum, modulo 998244353, of all integers between 1 and N (inclusive) that contain all of C_1, \ldots, C_M when written in base 10 without unnecessary leading zeros.

Constraints

• 1 \leq N < 10^{10^4}
• 1 \leq M \leq 10
• 0 \leq C_1 < \ldots < C_M \leq 9
• All values in input are integers.

Sample Input 1

104
2
0 1

Sample Output 1

520

Between 1 and 104, there are six integers that contain both 0 and 1 when written in base 10: 10,100,101,102,103,104.
The sum of them is 520.

Sample Input 2

999
4
1 2 3 4

Sample Output 2

0

Between 1 and 999, no integer contains all of 1, 2, 3, 4.

Sample Input 3

1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
5
0 2 4 6 8

Sample Output 3

397365274

Be sure to find the sum modulo 998244353.