Problem Statement

You are given a simple undirected graph with N vertices and M edges. The vertices are numbered 1 to N, and the i-th edge connects vertex A_i and vertex B_i. Let us delete zero or more edges to remove cycles from the graph. Find the minimum number of edges that must be deleted for this purpose.

Constraints

• 1 \leq N \leq 2 \times 10^5
• 0 \leq M \leq 2 \times 10^5
• 1 \leq A_i, B_i \leq N
• The given graph is simple.
• All values in the input are integers.

Sample Input 1

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

Sample Output 1

2

One way to remove cycles from the graph is to delete the two edges between vertex 1 and vertex 2 and between vertex 4 and vertex 5.
There is no way to remove cycles from the graph by deleting one or fewer edges, so you should print 2.

Sample Input 2

4 2
1 2
3 4

Sample Output 2

0

Sample Input 3

5 3
1 2
1 3
2 3

Sample Output 3

1

Problem Statement

You are given three sequences A=(A_1,\ldots,A_N), B=(B_1,\ldots,B_M), and C=(C_1,\ldots,C_L).

Additionally, a sequence X=(X_1,\ldots,X_Q) is given. For each i=1,\ldots,Q, solve the following problem:

Problem: Is it possible to select one element from each of A, B, and C so that their sum is X_i?

Constraints

• 1 \leq N,M,L \leq 100
• 0 \leq A_i, B_i ,C_i \leq 10^8
• 1 \leq Q \leq 2\times 10^5
• 0 \leq X_i \leq 3\times 10^8
• All input values are integers.

Sample Input 1

3
1 2 3
2
2 4
6
1 2 4 8 16 32
4
1 5 10 50

Sample Output 1

No
Yes
Yes
No

• It is impossible to select one element from each of A, B, and C so that their sum is 1.
• Selecting 1, 2, and 2 from A, B, and C, respectively, makes the sum 5.
• Selecting 2, 4, and 4 from A, B, and C, respectively, makes the sum 10.
• It is impossible to select one element from each of A, B, and C so that their sum is 50.

Problem Statement

Takahashi is playing a game.

The game consists of N stages numbered 1,2,\ldots,N. Initially, only stage 1 can be played.

For each stage i ( 1\leq i \leq N-1 ) that can be played, you can perform one of the following two actions at stage i:

• Spend A_i seconds to clear stage i. This allows you to play stage i+1.
• Spend B_i seconds to clear stage i. This allows you to play stage X_i.

Ignoring the times other than the time spent to clear the stages, how many seconds will it take at the minimum to be able to play stage N?

Constraints

• 2 \leq N \leq 2\times 10^5
• 1 \leq A_i, B_i \leq 10^9
• 1 \leq X_i \leq N
• All input values are integers.

Sample Input 1

5
100 200 3
50 10 1
100 200 5
150 1 2

Sample Output 1

350

By acting as follows, you will be allowed to play stage 5 in 350 seconds.

• Spend 100 seconds to clear stage 1, which allows you to play stage 2.
• Spend 50 seconds to clear stage 2, which allows you to play stage 3.
• Spend 200 seconds to clear stage 3, which allows you to play stage 5.

Sample Input 2

10
1000 10 9
1000 10 10
1000 10 2
1000 10 3
1000 10 4
1000 10 5
1000 10 6
1000 10 7
1000 10 8

Sample Output 2

90

Sample Input 3

6
1000000000 1000000000 1
1000000000 1000000000 1
1000000000 1000000000 1
1000000000 1000000000 1
1000000000 1000000000 1

Sample Output 3

5000000000

Problem Statement

Given is a positive integer N. Find the value \displaystyle\sum_{i=1}^N \left[ \frac{N}{i} \right].

Here, for a real number x, [x] denotes the largest integer not exceeding x.

Constraints

• 1 \leq N \leq 10^{12}
• N is an integer.

Sample Input 1

3

Sample Output 1

5

We have \left[ \frac{3}{1} \right]+\left[ \frac{3}{2} \right]+\left[ \frac{3}{3} \right]=3+1+1=5.

Sample Input 2

10000000000

Sample Output 2

231802823220

Note that the input and output may not fit into a 32-bit integer type.

Problem Statement

Takahashi is on an xy-plane. Initially, he is at point (s _ x,s _ y), and he wants to reach point (t _ x,t _ y).

On the xy-plane is a rectangle R\coloneqq\lbrace(x,y)\mid a-0.5\leq x\leq b+0.5,c-0.5\leq y\leq d+0.5\rbrace. Consider the following operation:

• Choose a lattice point (x,y) contained in the rectangle R. Takahashi teleports to the point symmetric to his current position with respect to point (x,y).

Determine if he can reach point (t _ x,t _ y) after repeating the operation above between 0 and 10^6 times, inclusive. If it is possible, construct a sequence of operations that leads him to point (t _ x,t _ y).

Constraints

• 0\leq s _ x,s _ y,t _ x,t _ y\leq2\times10^5
• 0\leq a\leq b\leq2\times10^5
• 0\leq c\leq d\leq2\times10^5
• All values in the input are integers.

Sample Input 1

1 2
7 8
7 9 0 3

Sample Output 1

Yes
7 0
9 3
7 1
8 1

For example, the following choices lead him from (1,2) to (7,8).

• Choose (7,0). Takahashi moves to (13,-2).
• Choose (9,3). Takahashi moves to (5,8).
• Choose (7,1). Takahashi moves to (9,-6).
• Choose (8,1). Takahashi moves to (7,8).

Any output that satisfies the conditions is accepted; for example, printing

Yes
7 3
9 0
7 2
9 1
8 1

is also accepted.

Sample Input 2

0 0
8 4
5 5 0 0

Sample Output 2

No

No sequence of operations leads him to point (8,4).

Sample Input 3

1 4
1 4
100 200 300 400

Sample Output 3

Yes

Takahashi may already be at the destination in the beginning.

Sample Input 4

22 2
16 7
14 30 11 14

Sample Output 4

No