Problem Statement

In an xy-coordinate plane whose x-axis is oriented to the right and whose y-axis is oriented upwards, rotate a point (a, b) around the origin d degrees counterclockwise and find the new coordinates of the point.

Constraints

• -1000 \leq a,b \leq 1000
• 1 \leq d \leq 360
• All values in input are integers.

Sample Input 1

2 2 180

Sample Output 1

-2 -2

When (2, 2) is rotated around the origin 180 degrees counterclockwise, it becomes the symmetric point of (2, 2) with respect to the origin, which is (-2, -2).

Sample Input 2

5 0 120

Sample Output 2

-2.49999999999999911182 4.33012701892219364908

When (5, 0) is rotated around the origin 120 degrees counterclockwise, it becomes (-\frac {5}{2} , \frac {5\sqrt{3}}{2}).
This sample output does not precisely match these values, but the errors are small enough to be considered correct.

Sample Input 3

0 0 11

Sample Output 3

0.00000000000000000000 0.00000000000000000000

Since (a, b) is the origin (the center of rotation), a rotation does not change its coordinates.

Sample Input 4

15 5 360

Sample Output 4

15.00000000000000177636 4.99999999999999555911

A 360-degree rotation does not change the coordinates of a point.

Sample Input 5

-505 191 278

Sample Output 5

118.85878514480690171240 526.66743699786547949770

Problem Statement

Print the color of the cell at the R-th row from the top and C-th column from the left in the following grid with 15 vertical rows and 15 horizontal columns.

Constraints

• 1 \leq R, C \leq 15
• R and C are integers.

Sample Input 1

3 5

Sample Output 1

black

In the grid above, the cell at the 3-rd row from the top and 5-th column from the left is black. Thus, black should be printed.

Sample Input 2

4 5

Sample Output 2

white

In the grid above, the cell at the 4-th row from the top and 5-th column from the left is white. Thus, white should be printed.

Problem Statement

N students are taking a 4-day exam.

There is a 300-point test on each day, for a total of 1200 points.

The first three days of the exam are already over, and the fourth day is now about to begin. The i-th student (1 \leq i \leq N) got P_{i, j} points on the j-th day (1 \leq j \leq 3).

For each student, determine whether it is possible that he/she is ranked in the top K after the fourth day.
Here, the rank of a student after the fourth day is defined as the number of students whose total scores over the four days are higher than that of the student, plus 1.

Constraints

• 1 \leq K \leq N \leq 10^5
• 0 \leq P_{i, j} \leq 300 \, (1 \leq i \leq N, 1 \leq j \leq 3)
• All values in input are integers.

Sample Input 1

3 1
178 205 132
112 220 96
36 64 20

Sample Output 1

Yes
Yes
No

If every student scores 100 on the fourth day, the 1-st student will rank 1-st.
If the 2-nd student scores 100 and the other students score 0 on the fourth day, the 2-nd student will rank 1-st.
The 3-rd student will never rank 1-st.

Sample Input 2

2 1
300 300 300
200 200 200

Sample Output 2

Yes
Yes

Sample Input 3

4 2
127 235 78
192 134 298
28 56 42
96 120 250

Sample Output 3

Yes
Yes
No
Yes

Problem Statement

We define sequences S_n as follows.

• S_1 is a sequence of length 1 containing a single 1.
• S_n (n is an integer greater than or equal to 2) is a sequence obtained by concatenating S_{n-1}, n, S_{n-1} in this order.

For example, S_2 and S_3 is defined as follows.

• S_2 is a concatenation of S_1, 2, and S_1, in this order, so it is 1,2,1.
• S_3 is a concatenation of S_2, 3, and S_2, in this order, so it is 1,2,1,3,1,2,1.

Given N, print the entire sequence S_N.

Constraints

• N is an integer.
• 1 \leq N \leq 16

Sample Input 1

2

Sample Output 1

1 2 1

As described in the Problem Statement, S_2 is 1,2,1.

Sample Input 2

1

Sample Output 2

1

Sample Input 3

4

Sample Output 3

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

• S_4 is a concatenation of S_3, 4, and S_3, in this order.

Problem Statement

The Republic of AtCoder has N cities numbered 1 through N and N-1 roads numbered 1 through N-1. Road i connects City A_i and City B_i bidirectionally. It is guaranteed that one can travel between every pair of cities using roads.

Takahashi will depart City 1 and have a journey by repeating the following.

• If there are unvisited cities that are directly connected to the city Takahashi is in now, he goes to the city with the smallest number among those cities.
• Otherwise,
  • if he is in City 1, he ends the journey;
  • otherwise, he goes to the city he was in just before visiting the current city for the first time.

Find the sequence of cities visited by Takahashi in the order he visits them.

Constraints

• 2 \leq N \leq 2\times 10^5
• 1\leq A_i,B_i \leq N
• It is possible to travel between every pair of cities using roads.

Sample Input 1

4
1 2
4 2
3 1

Sample Output 1

1 2 4 2 1 3 1

His journey will be as follows.

• Start in City 1.
• The unvisited cities directly connected to City 1 are City 2 and 3. Go to the city with the smaller number, that is, City 2.
• The unvisited city directly connected to City 2 is City 4. Go there.
• There is no unvisited city directly connected to City 4. Go back to City 2.
• There is no unvisited city directly connected to City 2. Go to City 1, where he was in just before visiting City 2 for the first time.
• The unvisited city directly connected to City 1 is City 3. Go there.
• There is no unvisited city directly connected to City 3. Go back to City 1.
• There is no unvisited city directly connected to City 1. End the journey.

Sample Input 2

5
1 2
1 3
1 4
1 5

Sample Output 2

1 2 1 3 1 4 1 5 1