Problem Statement

There is a rectangle in the xy-plane. Each edge of this rectangle is parallel to the x- or y-axis, and its area is not zero.

Given the coordinates of three of the four vertices of this rectangle, (x_1, y_1), (x_2, y_2), and (x_3, y_3), find the coordinates of the other vertex.

Constraints

• -100 \leq x_i, y_i \leq 100
• There uniquely exists a rectangle with all of (x_1, y_1), (x_2, y_2), (x_3, y_3) as vertices, edges parallel to the x- or y-axis, and a non-zero area.
• All values in input are integers.

Sample Input 1

-1 -1
-1 2
3 2

Sample Output 1

3 -1

The other vertex of the rectangle with vertices (-1, -1), (-1, 2), (3, 2) is (3, -1).

Sample Input 2

-60 -40
-60 -80
-20 -80

Sample Output 2

-20 -40

Problem Statement

A sport event is held in June of every year whose remainder when divided by 4 is 2.
Suppose that it is now January of the year Y. In what year will this sport event be held next time?

Constraints

• 2000 \leq Y \leq 3000
• Y is an integer.

Sample Input 1

2022

Sample Output 1

2022

The remainder when 2022 is divided by 4 is 2, so if it is now January of 2022, the next games will be held in June of the same year.

Sample Input 2

2023

Sample Output 2

2026

Sample Input 3

3000

Sample Output 3

3002

Problem Statement

You are given integer sequences, each of length N: A = (A_1, A_2, \dots, A_N) and B = (B_1, B_2, \dots, B_N).
All elements of A are different. All elements of B are different, too.

Print the following two values.

1. The number of integers contained in both A and B, appearing at the same position in the two sequences. In other words, the number of integers i such that A_i = B_i.
2. The number of integers contained in both A and B, appearing at different positions in the two sequences. In other words, the number of pairs of integers (i, j) such that A_i = B_j and i \neq j.

Constraints

• 1 \leq N \leq 1000
• 1 \leq A_i \leq 10^9
• 1 \leq B_i \leq 10^9
• A_1, A_2, \dots, A_N are all different.
• B_1, B_2, \dots, B_N are all different.
• All values in input are integers.

Sample Input 1

4
1 3 5 2
2 3 1 4

Sample Output 1

1
2

There is one integer contained in both A and B, appearing at the same position in the two sequences: A_2 = B_2 = 3.
There are two integers contained in both A and B, appearing at different positions in the two sequences: A_1 = B_3 = 1 and A_4 = B_1 = 2.

Sample Input 2

3
1 2 3
4 5 6

Sample Output 2

0
0

In both 1. and 2., no integer satisfies the condition.

Sample Input 3

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

Sample Output 3

3
2

Problem Statement

You are given a string S of length N consisting of lowercase English letters. The x-th (1 \le x \le N) character of S is S_x.

For each i=1,2,\dots,N-1, find the maximum non-negative integer l that satisfies all of the following conditions:

• l+i \le N, and
• for all integers k such that 1 \le k \le l, it holds that S_{k} \neq S_{k+i}.

Note that l=0 always satisfies the conditions.

Constraints

• N is an integer such that 2 \le N \le 5000.
• S is a string of length N consisting of lowercase English letters.

Sample Input 1

6
abcbac

Sample Output 1

5
1
2
0
1

In this input, S= abcbac.

• When i=1, we have S_1 \neq S_2, S_2 \neq S_3, \dots, and S_5 \neq S_6, so the maximum value is l=5.
• When i=2, we have S_1 \neq S_3 but S_2 = S_4, so the maximum value is l=1.
• When i=3, we have S_1 \neq S_4 and S_2 \neq S_5 but S_3 = S_6, so the maximum value is l=2.
• When i=4, we have S_1 = S_5, so the maximum value is l=0.
• When i=5, we have S_1 \neq S_6, so the maximum value is l=1.

Problem Statement

We have a grid with H horizontal rows and W vertical columns. We denote by (i, j) the cell at the i-th row from the top and j-th column from the left of the grid.
Each cell in the grid has a symbol # or . written on it. Let C[i][j] be the character written on (i, j). For integers i and j such that at least one of 1 \leq i \leq H and 1 \leq j \leq W is violated, we define C[i][j] to be ..

(4n+1) squares, consisting of (a, b) and (a+d,b+d),(a+d,b-d),(a-d,b+d),(a-d,b-d) (1 \leq d \leq n, 1 \leq n), are said to be a cross of size n centered at (a,b) if and only if all of the following conditions are satisfied:

• C[a][b] is #.
• C[a+d][b+d],C[a+d][b-d],C[a-d][b+d], and C[a-d][b-d] are all #, for all integers d such that 1 \leq d \leq n,
• At least one of C[a+n+1][b+n+1],C[a+n+1][b-n-1],C[a-n-1][b+n+1], and C[a-n-1][b-n-1] is ..

For example, the grid in the following figure has a cross of size 1 centered at (2, 2) and another of size 2 centered at (3, 7).

The grid has some crosses. No # is written on the cells except for those comprising a cross.
Additionally, no two squares that comprise two different crosses share a corner. The two grids in the following figure are the examples of grids where two squares that comprise different crosses share a corner; such grids are not given as an input. For example, the left grid is invalid because (3, 3) and (4, 4) share a corner.

Let N = \min(H, W), and S_n be the number of crosses of size n. Find S_1, S_2, \dots, S_N.

Constraints

• 3 \leq H, W \leq 100
• C[i][j] is # or ..
• No two different squares that comprise two different crosses share a corner.
• H and W are integers.

Sample Input 1

5 9
#.#.#...#
.#...#.#.
#.#...#..
.....#.#.
....#...#

Sample Output 1

1 1 0 0 0

As described in the Problem Statement, there are a cross of size 1 centered at (2, 2) and another of size 2 centered at (3, 7).

Sample Input 2

3 3
...
...
...

Sample Output 2

0 0 0

There may be no cross.

Sample Input 3

3 16
#.#.....#.#..#.#
.#.......#....#.
#.#.....#.#..#.#

Sample Output 3

3 0 0

Sample Input 4

15 20
#.#..#.............#
.#....#....#.#....#.
#.#....#....#....#..
........#..#.#..#...
#.....#..#.....#....
.#...#....#...#..#.#
..#.#......#.#....#.
...#........#....#.#
..#.#......#.#......
.#...#....#...#.....
#.....#..#.....#....
........#.......#...
#.#....#....#.#..#..
.#....#......#....#.
#.#..#......#.#....#

Sample Output 4

5 0 1 0 0 0 1 0 0 0 0 0 0 0 0

Problem Statement

Takahashi, who works for a pizza restaurant, is making a delicious cheese pizza for staff meals.
There are N kinds of cheese in front of him.
The deliciousness of the i-th kind of cheese is A_i per gram, and B_i grams of this cheese are available.
The deliciousness of the pizza will be the total deliciousness of cheese he puts on top of the pizza.
However, using too much cheese would make his boss angry, so the pizza can have at most W grams of cheese on top of it.
Under this condition, find the maximum possible deliciousness of the pizza.

Constraints

• All values in input are integers.
• 1 \le N \le 3 \times 10^5
• 1 \le W \le 3 \times 10^8
• 1 \le A_i \le 10^9
• 1 \le B_i \le 1000

Sample Input 1

3 5
3 1
4 2
2 3

Sample Output 1

15

The optimal choice is to use 1 gram of cheese of the first kind, 2 grams of the second kind, and 2 grams of the third kind.
The pizza will have a deliciousness of 15.

Sample Input 2

4 100
6 2
1 5
3 9
8 7

Sample Output 2

100

There may be less than W grams of cheese in total.

Sample Input 3

10 3141
314944731 649
140276783 228
578012421 809
878510647 519
925326537 943
337666726 611
879137070 306
87808915 39
756059990 244
228622672 291

Sample Output 3

2357689932073

Problem Statement

There are N cards with an integer written on each side. Card i (1 \leq i \leq N) has an integer a_i written on the front and an integer b_i written on the back.

You may choose whether to place each card with its front or back side visible.

Determine if you can place the cards so that the sum of the visible integers exactly equals S. If possible, find a placement of the cards to realize it.

Constraints

• 1 \leq N \leq 100
• 1 \leq S \leq 10000
• 1 \leq a_i, b_i \leq 100 \, (1 \leq i \leq N)
• All values in the input are integers.

Sample Input 1

3 11
1 4
2 3
5 7

Sample Output 1

Yes
THH

For example, the following placements make the sum of the visible integers exactly equal S (= 11):

• Place the 1-st card with its front side visible, 2-nd with its back, and 3-rd with its back.
• Place the 1-st card with its back side visible, 2-nd with its front, and 3-rd with its front.

Therefore, outputs like HTT and THH are accepted.

Sample Input 2

5 25
2 8
9 3
4 11
5 1
12 6

Sample Output 2

No

You cannot place the cards so that the sum of the visible integers exactly equals S (= 25).

Problem Statement

Takahashi is at the origin of a two-dimensional plane.
Takahashi will repeat teleporting N times. In each teleportation, he makes one of the following moves:

• Move from the current coordinates (x,y) to (x+A,y+B)
• Move from the current coordinates (x,y) to (x+C,y+D)
• Move from the current coordinates (x,y) to (x+E,y+F)

There are obstacles on M points (X_1,Y_1),\ldots,(X_M,Y_M) on the plane; he cannot teleport to these coordinates.

How many paths are there resulting from the N teleportations? Find the count modulo 998244353.

Constraints

• 1 \leq N \leq 300
• 0 \leq M \leq 10^5
• -10^9 \leq A,B,C,D,E,F \leq 10^9
• (A,B), (C,D), and (E,F) are distinct.
• -10^9 \leq X_i,Y_i \leq 10^9
• (X_i,Y_i)\neq(0,0)
• (X_i,Y_i) are distinct.
• All values in input are integers.

Sample Input 1

2 2
1 1 1 2 1 3
1 2
2 2

Sample Output 1

5

The following 5 paths are possible:

• (0,0)\to(1,1)\to(2,3)
• (0,0)\to(1,1)\to(2,4)
• (0,0)\to(1,3)\to(2,4)
• (0,0)\to(1,3)\to(2,5)
• (0,0)\to(1,3)\to(2,6)

Sample Input 2

10 3
-1000000000 -1000000000 1000000000 1000000000 -1000000000 1000000000
-1000000000 -1000000000
1000000000 1000000000
-1000000000 1000000000

Sample Output 2

0

Sample Input 3

300 0
0 0 1 0 0 1

Sample Output 3

292172978

Problem Statement

Given are sequences of N integers each: A = (A_1, \dots, A_N) and B = (B_1, \dots, B_N). Find the number of non-empty subsets S of \{1,2,\ldots,N\} that satisfy the following condition:

• \max_{i \in S} A_i \geq \sum_{i \in S} B_i.

Since the count can be enormous, print it modulo 998244353.

Constraints

• 1 \leq N \leq 5000
• 1 \leq A_i,B_i \leq 5000
• All values in input are integers.

Sample Input 1

2
3 1
1 2

Sample Output 1

2

\{1,2,\ldots,N\} has three subsets: \{1\}, \{2\}, and \{1,2\}.

• For S=\{1\}, we have \max_{i \in S} A_i=3 and \sum_{i \in S} B_i=1.
• For S=\{2\}, we have \max_{i \in S} A_i=1 and \sum_{i \in S} B_i=2.
• For S=\{1,2\}, we have \max_{i \in S} A_i=3 and \sum_{i \in S} B_i=3.

Thus, the condition \max_{i \in S} A_i \geq \sum_{i \in S} B_i is satisfied by two subsets: \{1\} and \{1,2\}.

Sample Input 2

2
1 1
2 2

Sample Output 2

0

There may be no subsets that satisfy the condition.

Sample Input 3

20
1937 3980 2689 1208 3640 1979 581 2271 4229 3948 3708 1522 4161 4661 3797 96 3388 3395 2920 2247
4485 2580 174 1156 3770 3396 3558 3500 3494 479 269 3383 1230 1711 3545 3919 134 475 3796 1017

Sample Output 3

476