Problem Statement

Takahashi, a young baseball enthusiast, has been a very good boy this year, so Santa has decided to give him a bat or a glove, whichever is more expensive.

If a bat costs B yen and a glove costs G yen (B\neq G), which one will Santa give to Takahashi?

Constraints

• B and G are different integers between 1 and 1000, inclusive.

Sample Input 1

300 100

Sample Output 1

Bat

The bat is more expensive than the glove, so Santa will give Takahashi the bat.

Sample Input 2

334 343

Sample Output 2

Glove

The glove is more expensive than the bat, so Santa will give Takahashi the glove.

Problem Statement

There is a road that stretches infinitely to the east and west, and the coordinate of a point located x meters to the east from a certain reference point on this road is defined as x. In particular, the coordinate of a point located x meters to the west from the reference point is -x.

Snuke will set up Christmas trees at points on the road at intervals of M meters, starting from a point with coordinate A. In other words, he will set up a Christmas tree at each point that can be expressed as A+kM using some integer k.

Takahashi and Aoki are standing at points with coordinates L and R (L\leq R), respectively. Find the number of Christmas trees that will be set up between Takahashi and Aoki (including the points where they are standing).

Constraints

• -10^{18}\leq A \leq 10^{18}
• 1\leq M \leq 10^9
• -10^{18}\leq L\leq R \leq 10^{18}
• All input values are integers.

Sample Input 1

5 3 -1 6

Sample Output 1

3

Snuke will set up Christmas trees at points with coordinates \dots,-4,-1,2,5,8,11,14\dots. Three of them at coordinates -1, 2, and 5 are between Takahashi and Aoki.

Sample Input 2

-2 2 1 1

Sample Output 2

0

Sometimes, Takahashi and Aoki are standing at the same point.

Sample Input 3

-177018739841739480 2436426 -80154573737296504 585335723211047198

Sample Output 3

273142010859

Problem Statement

Takahashi has N pairs of socks, and the i-th pair consists of two socks of color i. One day, after organizing his chest of drawers, Takahashi realized that he had lost one sock each of colors A_1, A_2, \dots, A_K, so he decided to use the remaining 2N-K socks to make \lfloor\frac{2N-K}{2}\rfloor new pairs of socks, each pair consisting of two socks. The weirdness of a pair of a sock of color i and a sock of color j is defined as |i-j|, and Takahashi wants to minimize the total weirdness.

Find the minimum possible total weirdness when making \lfloor\frac{2N-K}{2}\rfloor pairs from the remaining socks. Note that if 2N-K is odd, there will be one sock that is not included in any pair.

Constraints

• 1\leq K\leq N \leq 2\times 10^5
• 1\leq A_1 < A_2 < \dots < A_K \leq N
• All input values are integers.

Sample Input 1

4 2
1 3

Sample Output 1

2

Below, let (i,j) denote a pair of a sock of color i and a sock of color j.

There are 1, 2, 1, 2 socks of colors 1, 2, 3, 4, respectively. Creating the pairs (1,2),(2,3),(4,4) results in a total weirdness of |1-2|+|2-3|+|4-4|=2, which is the minimum.

Sample Input 2

5 1
2

Sample Output 2

0

The optimal solution is to make the pairs (1,1),(3,3),(4,4),(5,5) and leave one sock of color 2 as a surplus (not included in any pair).

Sample Input 3

8 5
1 2 4 7 8

Sample Output 3

2

Problem Statement

There are N sleighs numbered 1,2,\ldots, N.

R_i reindeer are required to pull sleigh i.

Additionally, each reindeer can pull at most one sleigh. More precisely, \sum_{k=1}^{m} R_{i_k} reindeer are required to pull m sleighs i_1, i_2, \ldots, i_m.

Find the answer to Q queries of the following form:

• You are given an integer X. Determine the maximum number of sleighs that can be pulled when there are X reindeer.

Constraints

• 1 \leq N, Q \leq 2 \times 10^5
• 1 \leq R_i \leq 10^9
• 1 \leq X \leq 2 \times 10^{14}
• All input values are integers.

Sample Input 1

4 3
5 3 11 8
16
7
1000

Sample Output 1

3
1
4

When there are 16 reindeer, sleighs 1,2,4 can be pulled.

It is impossible to pull four sleighs with 16 reindeer, so the answer to query 1 is 3.

Sample Input 2

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

Sample Output 2

1
1
2
2
2
3

Sample Input 3

2 2
1000000000 1000000000
200000000000000
1

Sample Output 3

2
0

Problem Statement

This problem has a similar setting to Problem G. Differences in the problem statement are indicated in red.

There is a grid with H rows and W columns, where each cell is painted red or green.

Let (i,j) denote the cell in the i-th row from the top and the j-th column from the left.

The color of cell (i,j) is represented by the character S_{i,j}, where S_{i,j} = . means cell (i,j) is red, and S_{i,j} = # means cell (i,j) is green.

The number of green connected components in the grid is the number of connected components in the graph with the vertex set being the green cells and the edge set being the edges connecting two adjacent green cells. Here, two cells (x,y) and (x',y') are considered adjacent when |x-x'| + |y-y'| = 1.

Consider choosing one red cell uniformly at random and repainting it green. Print the expected value of the number of green connected components in the grid after repainting, modulo 998244353.

Constraints

• 1 \leq H,W \leq 1000
• S_{i,j} = . or S_{i,j} = #.
• There is at least one (i,j) such that S_{i,j} = ..

Sample Input 1

3 3
##.
#.#
#..

Sample Output 1

499122178

If cell (1,3) is repainted green, the number of green connected components becomes 1.

If cell (2,2) is repainted green, the number of green connected components becomes 1.

If cell (3,2) is repainted green, the number of green connected components becomes 2.

If cell (3,3) is repainted green, the number of green connected components becomes 2.

Therefore, the expected value of the number of green connected components after choosing one red cell uniformly at random and repainting it green is (1+1+2+2)/4 = 3/2.

Sample Input 2

4 5
..#..
.###.
#####
..#..

Sample Output 2

598946613

Sample Input 3

3 4
#...
.#.#
..##

Sample Output 3

285212675

Problem Statement

There is a town represented as an xy-plane, where Santa lives, along with N children numbered 1 to N. Santa's house is at coordinates (S_X,S_Y), and the house of child i\ (1\leq i\leq N) is at (X_i,Y_i).

Santa wants to deliver one present to each of the N children in numerical order. To deliver a present to child i, Santa must visit the house of child i with at least one present in hand. However, Santa can only carry up to K presents at a time, and he must return to his own house to replenish presents (there are enough presents at Santa's house).

Find the minimum distance Santa must travel to leave his house, deliver presents to all N children, and return to his house.

Constraints

• 1\leq K\leq N \leq 2\times 10^5
• -10^9\leq S_X,S_Y,X_i,Y_i \leq 10^9
• (S_X,S_Y)\neq (X_i,Y_i)
• (X_i,Y_i)\neq (X_j,Y_j)\ (i\neq j)
• All input values are integers.

Sample Input 1

3 2
1 1
3 1
1 2
3 2

Sample Output 1

9.236067977499790

In the figure above, the red circle represents Santa's house, and the circles with numbers represent the houses of the children with those numbers.

Consider Santa acting as follows:

1. Leave his house with two presents.
2. Go to child 1's house and deliver a present.
3. Return to his house and replenish one present.
4. Go to child 2's house and deliver a present.
5. Go to child 3's house and deliver a present.
6. Return to his house.

In this case, Santa travels the distance of 2+2+1+2+\sqrt{5}=7+\sqrt{5}=9.236\dots, which is the minimum.

Sample Input 2

2 1
0 1
-1 1
1 1

Sample Output 2

4.000000000000000

Sample Input 3

8 3
735867677 193944314
586260100 -192321079
95834122 802780784
418379342 -790013317
-445130206 189801569
-354684803 -49687658
-204491568 -840249197
853829789 470958158
-751917965 762048217

Sample Output 3

11347715738.116592407226562

Problem Statement

This problem has a similar setting to Problem E. Differences in the problem statement are indicated in red.

There is a grid with H rows and W columns, where each cell is painted red or green.

Let (i,j) denote the cell in the i-th row from the top and the j-th column from the left.

The color of cell (i,j) is represented by the character S_{i,j}, where S_{i,j} = . means cell (i,j) is red, and S_{i,j} = # means cell (i,j) is green.

The number of green connected components in the grid is the number of connected components in the graph with the vertex set being the green cells and the edge set being the edges connecting two adjacent green cells. Here, two cells (x,y) and (x',y') are considered adjacent when |x-x'| + |y-y'| = 1.

Consider choosing one green cell uniformly at random and repainting it red. Print the expected value of the number of green connected components in the grid after repainting, modulo 998244353.

Constraints

• 1 \leq H,W \leq 1000
• S_{i,j} = . or S_{i,j} = #.
• There is at least one (i,j) such that S_{i,j} = #.

Sample Input 1

3 3
##.
#.#
#..

Sample Output 1

598946614

If cell (1,1) is repainted red, the number of green connected components becomes 3.

If cell (1,2) is repainted red, the number of green connected components becomes 2.

If cell (2,1) is repainted red, the number of green connected components becomes 3.

If cell (2,3) is repainted red, the number of green connected components becomes 1.

If cell (3,1) is repainted red, the number of green connected components becomes 2.

Therefore, the expected value of the number of green connected components after choosing one green cell uniformly at random and repainting it red is (3+2+3+1+2)/5 =11/5.

Sample Input 2

4 5
..#..
.###.
#####
..#..

Sample Output 2

199648872

Sample Input 3

3 4
#...
.#.#
..##

Sample Output 3

399297744