Problem Statement

There is a single integer N written on a blackboard.
Takahashi will repeat the following series of operations until all integers not less than 2 are removed from the blackboard:

• Choose one integer x not less than 2 written on the blackboard.
• Erase one occurrence of x from the blackboard. Then, write two new integers \left \lfloor \dfrac{x}{2} \right\rfloor and \left\lceil \dfrac{x}{2} \right\rceil on the blackboard.
• Takahashi must pay x yen to perform this series of operations.

Here, \lfloor a \rfloor denotes the largest integer not greater than a, and \lceil a \rceil denotes the smallest integer not less than a.

What is the total amount of money Takahashi will have paid when no more operations can be performed?
It can be proved that the total amount he will pay is constant regardless of the order in which the operations are performed.

Constraints

• 2 \leq N \leq 10^{17}

Sample Input 1

3

Sample Output 1

5

Here is an example of how Takahashi performs the operations:

• Initially, there is one 3 written on the blackboard.
• He chooses 3. He pays 3 yen, erases one 3 from the blackboard, and writes \left \lfloor \dfrac{3}{2} \right\rfloor = 1 and \left\lceil \dfrac{3}{2} \right\rceil = 2 on the blackboard.
• There is one 2 and one 1 written on the blackboard.
• He chooses 2. He pays 2 yen, erases one 2 from the blackboard, and writes \left \lfloor \dfrac{2}{2} \right\rfloor = 1 and \left\lceil \dfrac{2}{2} \right\rceil = 1 on the blackboard.
• There are three 1s written on the blackboard.
• Since all integers not less than 2 have been removed from the blackboard, the process is finished.

Takahashi has paid a total of 3 + 2 = 5 yen for the entire process, so print 5.

Sample Input 2

340

Sample Output 2

2888

Sample Input 3

100000000000000000

Sample Output 3

5655884811924144128

Problem Statement

There are N people numbered 1, 2, \ldots, N on a two-dimensional plane, and person i is at the point represented by the coordinates (X_i,Y_i).

Person 1 has been infected with a virus. The virus spreads to people within a distance of D from an infected person.

Here, the distance is defined as the Euclidean distance, that is, for two points (a_1, a_2) and (b_1, b_2), the distance between these two points is \sqrt {(a_1-b_1)^2 + (a_2-b_2)^2}.

After a sufficient amount of time has passed, that is, when all people within a distance of D from person i are infected with the virus if person i is infected, determine whether person i is infected with the virus for each i.

Constraints

• 1 \leq N, D \leq 2000
• -1000 \leq X_i, Y_i \leq 1000
• (X_i, Y_i) \neq (X_j, Y_j) if i \neq j.
• All input values are integers.

Sample Input 1

4 5
2 -1
3 1
8 8
0 5

Sample Output 1

Yes
Yes
No
Yes

The distance between person 1 and person 2 is \sqrt 5, so person 2 gets infected with the virus.
Also, the distance between person 2 and person 4 is 5, so person 4 gets infected with the virus.
Person 3 has no one within a distance of 5, so they will not be infected with the virus.

Sample Input 2

3 1
0 0
-1000 -1000
1000 1000

Sample Output 2

Yes
No
No

Sample Input 3

9 4
3 2
6 -1
1 6
6 5
-2 -3
5 3
2 -3
2 1
2 6

Sample Output 3

Yes
No
No
Yes
Yes
Yes
Yes
Yes
No

Problem Statement

There is an N \times N grid. Takahashi wants to color each cell black or white so that all of the following conditions are satisfied:

• For every row, the following condition holds:
• There exists an integer i\ (0\leq i\leq N) such that the leftmost i cells are colored black, and the rest are colored white.
• For every column, the following condition holds:
• There exists an integer i\ (0\leq i\leq N) such that the topmost i cells are colored black, and the rest are colored white.

Out of these N^2 cells, M of them have already been colored. Among them, the i-th one is at the X_i-th row from the top and the Y_i-th column from the left, and it is colored black if C_i is B and white if C_i is W.

Determine whether he can color the remaining uncolored N^2 - M cells so that all the conditions are satisfied.

Constraints

• 1\leq N\leq 10^9
• 1\leq M\leq \min(N^2,2\times 10^5)
• 1\leq X_i,Y_i\leq N
• (X_i,Y_i)\neq (X_j,Y_j)\ (i\neq j)
• C_i is B or W.
• All input numbers are integers.

Sample Input 1

4 3
4 1 B
3 2 W
1 3 B

Sample Output 1

Yes

For example, one can color the grid as in the following figure to satisfy the conditions. The cells already colored are surrounded by red borders.

Sample Input 2

2 2
1 2 W
2 2 B

Sample Output 2

No

No matter how the remaining two cells are colored, the conditions cannot be satisfied.

Sample Input 3

1 1
1 1 W

Sample Output 3

Yes

Sample Input 4

2289 10
1700 1083 W
528 967 B
1789 211 W
518 1708 W
1036 779 B
136 657 B
759 1497 B
902 1309 B
1814 712 B
936 763 B

Sample Output 4

No

Problem Statement

There are N people gathered for an event called Flowing Noodles. The people are lined up in a row, numbered 1 to N in order from front to back.

During the event, the following occurrence happens M times:

• At time T_i, a quantity W_i of noodles is flown down. The person at the front of the row gets all of it (if no one is in the row, no one gets it). That person then steps out of the row and returns to their original position in the row at time T_i+S_i.

A person who returns to the row at time X is considered to be in the row at time X.

After all the M occurrences, report the total amount of noodles each person has got.

Constraints

• 1 \leq N \leq 2\times 10^5
• 1 \leq M \leq 2\times 10^5
• 0 <T_1 <\ldots < T_M \leq 10^9
• 1 \leq S_i \leq 10^9
• 1 \leq W_i \leq 10^9
• All input values are integers.

Sample Input 1

3 5
1 1 3
2 10 100
4 100 10000
10 1000 1000000000
100 1000000000 1

Sample Output 1

101
10
1000

The event proceeds as follows:

• At time 1, a quantity 1 of noodles is flown down. People 1, 2, and 3 are in the row, and the person at the front, person 1, gets the noodles and steps out of the row.
• At time 2, a quantity 10 of noodles is flown down. People 2 and 3 are in the row, and the person at the front, person 2, gets the noodles and steps out of the row.
• At time 4, person 1 returns to the row.
• At time 4, a quantity 100 of noodles is flown down. People 1 and 3 are in the row, and the person at the front, person 1, gets the noodles and steps out of the row.
• At time 10, a quantity 1000 of noodles is flown down. Only person 3 is in the row, and the person at the front, person 3, gets the noodles and steps out of the row.
• At time 100, a quantity 1000000000 of noodles is flown down. No one is in the row, so no one gets these noodles.
• At time 102, person 2 returns to the row.
• At time 10004, person 1 returns to the row.
• At time 1000000010, person 3 returns to the row.

The total amounts of noodles people 1, 2, and 3 have got are 101, 10, and 1000, respectively.

Sample Input 2

3 1
1 1 1

Sample Output 2

1
0
0

Sample Input 3

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

Sample Output 3

15

Problem Statement

You are given a tree with N vertices. The i-th edge (1 \leq i \leq N-1) connects vertices u_i and v_i bidirectionally.

Adding one undirected edge to the given tree always yields a graph with exactly one cycle.

Among such graphs, how many satisfy all of the following conditions?

• The graph is simple.
• All vertices in the cycle have degree 3.

Constraints

• 3 \leq N \leq 2 \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

6
1 2
2 3
3 4
4 5
3 6

Sample Output 1

1

Adding an edge connecting vertices 2 and 4 yields a simple graph where all vertices in the cycle have degree 3, so it satisfies the conditions.

Sample Input 2

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

Sample Output 2

0

There are cases where no graphs satisfy the conditions.

Sample Input 3

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

Sample Output 3

6