Problem Statement

AtCoder Regular Contest (ARC) currently has two divisions: Div. 1 and Div. 2. In ARC Div. 1, participants whose rating is between 1600 and 2999, inclusive, are rated. In ARC Div. 2, participants whose rating is between 1200 and 2399, inclusive, are rated.

You are given positive integers R and X.

Determine whether a person with rating R is rated in ARC Div. X.

Constraints

• 1 \le R \le 4229
• 1 \le X \le 2
• All input values are integers.

Sample Input 1

2000 1

Sample Output 1

Yes

Because 2000 lies between 1600 and 2999, a person with rating 2000 is rated in ARC Div. 1.

Sample Input 2

1000 1

Sample Output 2

No

Because 1000 is less than 1600, a person with rating 1000 is not rated in ARC Div. 1.

Sample Input 3

1500 2

Sample Output 3

Yes

Because 1500 lies between 1200 and 2399, a person with rating 1500 is rated in ARC Div. 2.

Sample Input 4

2800 2

Sample Output 4

No

Because 2800 exceeds 2399, a person with rating 2800 is not rated in ARC Div. 2.

Problem Statement

In the hexadecimal system, where the digits ABCDEF corresponding to 10,11,12,13,14, and 15 are used in addition to 0123456789, every integer between 0 and 255 is represented as a 1- or 2-digit numeral.
For example, 0 and 12 are represented as 1-digit hexadecimal numerals 0 and C; 99 and 255 are represented as 2-digit hexadecimals 63 and FF.

Given an integer N between 0 and 255, convert it to an exactly two-digit hexadecimal numeral, prepending leading 0s if necessary.

Notes

The judge is case-sensitive. Specifically, you cannot use abcdef as hexadecimal digits instead of ABCDEF.

Constraints

• 0 \leq N \leq 255
• N is an integer.

Sample Input 1

99

Sample Output 1

63

99 is represented as 63 in hexadecimal.

Sample Input 2

12

Sample Output 2

0C

12 is represented as C in hexadecimal.
Since we ask you to convert it to a two-digit hexadecimal numeral, the answer is 0C, where 0 is prepended to C.

Sample Input 3

0

Sample Output 3

00

Sample Input 4

255

Sample Output 4

FF

Problem Statement

There are A slimes.

Each time Snuke shouts, the slimes multiply by K times.

In order to have B or more slimes, at least how many times does Snuke need to shout?

Constraints

• 1 \leq A \leq B \leq 10^9
• 2 \leq K \leq 10^9
• All values in input are integers.

Sample Input 1

1 4 2

Sample Output 1

2

We start with one slime. After Snuke's first shout, we have two slimes; after his second shout, we have four slimes. Thus, he needs to shout at least twice to have four or more slimes.

Sample Input 2

7 7 10

Sample Output 2

0

We have seven slimes already at the start.

Sample Input 3

31 415926 5

Sample Output 3

6

Problem Statement

In the xy-plane, there are three points A(x_A, y_A), B(x_B, y_B), and C(x_C, y_C) that are not collinear. Determine whether the triangle ABC is a right triangle.

Constraints

• -1000 \leq x_A, y_A, x_B, y_B, x_C, y_C \leq 1000
• The three points A, B, and C are not collinear.
• All input values are integers.

Sample Input 1

0 0
4 0
0 3

Sample Output 1

Yes

The triangle ABC is a right triangle.

Sample Input 2

-4 3
2 1
3 4

Sample Output 2

Yes

The triangle ABC is a right triangle.

Sample Input 3

2 4
-3 2
1 -2

Sample Output 3

No

The triangle ABC is not a right triangle.

Problem Statement

We have N cups of ice cream.
The flavor and deliciousness of the i-th cup are F_i and S_i, respectively (S_i is an even number).

You will choose and eat two of the N cups.
Your satisfaction here is defined as follows.

• Let s and t (s \ge t) be the deliciousness of the eaten cups.
  • If the two cups have different flavors, your satisfaction is \displaystyle s+t.
  • Otherwise, your satisfaction is \displaystyle s + \frac{t}{2}.

Find the maximum achievable satisfaction.

Constraints

• All input values are integers.
• 2 \le N \le 3 \times 10^5
• 1 \le F_i \le N
• 2 \le S_i \le 10^9
• S_i is even.

Sample Input 1

4
1 4
2 10
2 8
3 6

Sample Output 1

16

Consider eating the second and fourth cups.

• The second cup has a flavor of 2 and deliciousness of 10.
• The fourth cup has a flavor of 3 and deliciousness of 6.
• Since they have different flavors, your satisfaction is 10+6=16.

Thus, you can achieve the satisfaction of 16.
You cannot achieve a satisfaction greater than 16.

Sample Input 2

4
4 10
3 2
2 4
4 12

Sample Output 2

17

Consider eating the first and fourth cups.

• The first cup has a flavor of 4 and deliciousness of 10.
• The fourth cup has a flavor of 4 and deliciousness of 12.
• Since they have the same flavor, your satisfaction is 12+\frac{10}{2}=17.

Thus, you can achieve the satisfaction of 17.
You cannot achieve a satisfaction greater than 17.

Problem Statement

There is a grid with H rows and W columns. Let (i, j) denote the square at the i-th row from the top and the j-th column from the left.
Initially, there was one cookie on each square inside a rectangle whose height and width were at least 2 squares long, and no cookie on the other squares.
Formally, there was exactly one quadruple of integers (a,b,c,d) that satisfied all of the following conditions.

• 1 \leq a \lt b \leq H
• 1 \leq c \lt d \leq W
• There was one cookie on each square (i, j) such that a \leq i \leq b, c \leq j \leq d, and no cookie on the other squares.

However, Snuke took and ate one of the cookies on the grid.
The square that contained that cookie is now empty.

As the input, you are given the state of the grid after Snuke ate the cookie.
The state of the square (i, j) is given as the character S_{i,j}, where # means a square with a cookie, and . means a square without one.
Find the square that contained the cookie eaten by Snuke. (The answer is uniquely determined.)

Constraints

• 2 \leq H, W \leq 500
• S_{i,j} is # or ..

Sample Input 1

5 6
......
..#.#.
..###.
..###.
......

Sample Output 1

2 4

Initially, cookies were on the squares inside the rectangle with (2, 3) as the top-left corner and (4, 5) as the bottom-right corner, and Snuke ate the cookie on (2, 4). Thus, you should print (2, 4).

Sample Input 2

3 2
#.
##
##

Sample Output 2

1 2

Initially, cookies were placed on the squares inside the rectangle with (1, 1) as the top-left corner and (3, 2) as the bottom-right corner, and Snuke ate the cookie at (1, 2).

Sample Input 3

6 6
..####
..##.#
..####
..####
..####
......

Sample Output 3

2 5

Problem Statement

You are given a sequence A=(A_1,A_2,\dots,A_N) of length N.

Process Q queries in order. Each query is in one of the following formats:

• 1 x : Swap the values of A_x and A_{x+1}.
• 2 l r : Find the value of \displaystyle \sum_{l\leq i\leq r} A_i.

Constraints

• 2\leq N \leq 2\times 10^5
• 1\leq Q \leq 5\times 10^5
• 1\leq A_i \leq 10^4
• For queries of the first type, 1\leq x \leq N-1.
• For queries of the second type, 1\leq l\leq r \leq N.
• All input values are integers.

Sample Input 1

4 4
2 7 1 8
1 2
2 1 2
1 1
2 2 4

Sample Output 1

3
17

• In the 1-st query, swap the values of A_2 and A_3. This makes A=(2,1,7,8).
• In the 2-nd query, find the value of A_1+A_2. The answer is 2+1=3.
• In the 3-rd query, swap the values of A_1 and A_2. This makes A=(1,2,7,8).
• In the 4-th query, find the value of A_2+A_3+A_4. The answer is 2+7+8=17.

Sample Input 2

8 10
22 75 26 45 72 81 47 29
2 2 7
2 6 8
2 4 4
1 2
2 1 3
1 1
2 2 4
1 2
1 4
2 1 1

Sample Output 2

346
157
45
123
142
26

Problem Statement

A "Snowflake Tree" is defined as a tree that can be generated by the following procedure:

1. Choose positive integers x,y.
2. Prepare one vertex.
3. Prepare x more vertices, and connect each of them to the vertex prepared in step 2.
4. For each of the x vertices prepared in step 3, attach y leaves to it.

The figure below shows a Snowflake Tree with x=4,y=2. The vertices prepared in steps 2, 3, 4 are shown in red, blue, and green, respectively.

You are given a tree T with N vertices. The vertices are numbered 1 to N, and the i-th edge (i=1,2,\dots,N-1) connects vertices u_i and v_i.

Consider deleting zero or more vertices of T and the edges adjacent to them so that the remaining graph becomes a single Snowflake Tree. Find the minimum number of vertices that must be deleted. Under the constraints of this problem, it is always possible to transform T into a Snowflake Tree.

Constraints

• 3 \leq N \leq 3 \times 10^5
• 1 \leq u_i < v_i \leq N
• The given graph is a tree.
• All input values are integers.

Sample Input 1

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

Sample Output 1

1

By deleting vertex 8, the given tree can be transformed into a Snowflake Tree with x=2,y=2.

Sample Input 2

3
1 2
2 3

Sample Output 2

0

The given tree is already a Snowflake Tree with x=1,y=1.

Sample Input 3

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

Sample Output 3

3

Problem Statement

The Republic of Atcoder has N towns numbered 1 through N, and M highways numbered 1 through M.
Highway i connects Town A_i and Town B_i bidirectionally.

King Takahashi is going to construct (N-M-1) new highways so that the following two conditions are satisfied:

• One can travel between every pair of towns using some number of highways
• For each i=1,\ldots,N, exactly D_i highways are directly connected to Town i

Determine if there is such a way of construction. If it exists, print one.

Constraints

• 2 \leq N \leq 2\times 10^5
• 0 \leq M \lt N-1
• 1 \leq D_i \leq N-1
• 1\leq A_i \lt B_i \leq N
• If i\neq j, then (A_i, B_i) \neq (A_j,B_j).
• All values in input are integers.

Sample Input 1

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

Sample Output 1

6 2
5 6
4 5

As in the Sample Output, the conditions can be satisfied by constructing highways connecting Towns 6 and 2, Towns 5 and 6, and Towns 4 and 5, respectively.

Another example to satisfy the conditions is to construct highways connecting Towns 6 and 4, Towns 5 and 6, and Towns 2 and 5, respectively.

Sample Input 2

5 1
1 1 1 1 4
2 3

Sample Output 2

-1

Sample Input 3

4 0
3 3 3 3

Sample Output 3

-1