Problem Statement

There is a 10^{100} \times 7 matrix A, where the (i,j)-th entry is (i-1) \times 7 + j for every pair of integers (i,j)\ (1 \leq i \leq 10^{100}, 1 \leq j \leq 7).

Given an N \times M matrix B, determine whether B is some (unrotated) rectangular part of A.

Constraints

• 1 \leq N \leq 10^4
• 1 \leq M \leq 7
• 1 \leq B_{i,j} \leq 10^9
• All values in input are integers.

Sample Input 1

2 3
1 2 3
8 9 10

Sample Output 1

Yes

The given matrix B is the top-left 2 \times 3 submatrix of A.

Sample Input 2

2 1
1
2

Sample Output 2

No

Although the given matrix B would match the top-left 1 \times 2 submatrix of A after rotating 90 degrees, the Problem Statement asks whether B is an unrotated part of A, so the answer is No.

Sample Input 3

10 4
1346 1347 1348 1349
1353 1354 1355 1356
1360 1361 1362 1363
1367 1368 1369 1370
1374 1375 1376 1377
1381 1382 1383 1384
1388 1389 1390 1391
1395 1396 1397 1398
1402 1403 1404 1405
1409 1410 1411 1412

Sample Output 3

Yes

Problem Statement

Takahashi, who works for a pizza restaurant, is making a delicious cheese pizza for staff meals.
There are N kinds of cheese in front of him.
The deliciousness of the i-th kind of cheese is A_i per gram, and B_i grams of this cheese are available.
The deliciousness of the pizza will be the total deliciousness of cheese he puts on top of the pizza.
However, using too much cheese would make his boss angry, so the pizza can have at most W grams of cheese on top of it.
Under this condition, find the maximum possible deliciousness of the pizza.

Constraints

• All values in input are integers.
• 1 \le N \le 3 \times 10^5
• 1 \le W \le 3 \times 10^8
• 1 \le A_i \le 10^9
• 1 \le B_i \le 1000

Sample Input 1

3 5
3 1
4 2
2 3

Sample Output 1

15

The optimal choice is to use 1 gram of cheese of the first kind, 2 grams of the second kind, and 2 grams of the third kind.
The pizza will have a deliciousness of 15.

Sample Input 2

4 100
6 2
1 5
3 9
8 7

Sample Output 2

100

There may be less than W grams of cheese in total.

Sample Input 3

10 3141
314944731 649
140276783 228
578012421 809
878510647 519
925326537 943
337666726 611
879137070 306
87808915 39
756059990 244
228622672 291

Sample Output 3

2357689932073

Problem Statement

We have a positive integer a. Additionally, there is a blackboard with a number written in base 10.
Let x be the number on the blackboard. Takahashi can do the operations below to change this number.

• Erase x and write x multiplied by a, in base 10.
• See x as a string and move the rightmost digit to the beginning.
  This operation can only be done when x \geq 10 and x is not divisible by 10.

For example, when a = 2, x = 123, Takahashi can do one of the following.

• Erase x and write x \times a = 123 \times 2 = 246.
• See x as a string and move the rightmost digit 3 of 123 to the beginning, changing the number from 123 to 312.

The number on the blackboard is initially 1. What is the minimum number of operations needed to change the number on the blackboard to N? If there is no way to change the number to N, print -1.

Constraints

• 2 \leq a \lt 10^6
• 2 \leq N \lt 10^6
• All values in input are integers.

Sample Input 1

3 72

Sample Output 1

4

We can change the number on the blackboard from 1 to 72 in four operations, as follows.

• Do the operation of the first type: 1 \to 3.
• Do the operation of the first type: 3 \to 9.
• Do the operation of the first type: 9 \to 27.
• Do the operation of the second type: 27 \to 72.

It is impossible to reach 72 in three or fewer operations, so the answer is 4.

Sample Input 2

2 5

Sample Output 2

-1

It is impossible to change the number on the blackboard to 5.

Sample Input 3

2 611

Sample Output 3

12

There is a way to change the number on the blackboard to 611 in 12 operations: 1 \to 2 \to 4 \to 8 \to 16 \to 32 \to 64 \to 46 \to 92 \to 29 \to 58 \to 116 \to 611, which is the minimum possible.

Sample Input 4

2 767090

Sample Output 4

111

Problem Statement

We have an empty sequence A. You will be given Q queries, which should be processed in the order they are given. Each query is of one of the three kinds below:

• 1 x : Append x to the end of A.
• 2 : Print the element at the beginning of A. Then, delete that element. It is guaranteed that A will not empty when this query is given.
• 3 : Sort A in ascending order.

Constraints

• 1 \leq Q \leq 2 \times 10^5
• 0 \leq x \leq 10^9
• A will not be empty when a query 2 is given.
• All values in input are integers.

Sample Input 1

8
1 4
1 3
1 2
1 1
3
2
1 0
2

Sample Output 1

1
2

The i-th line below shows the contents of A after the i-th query is processed in Sample Input 1.

• (4)
• (4, 3)
• (4, 3, 2)
• (4, 3, 2, 1)
• (1, 2, 3, 4)
• (2, 3, 4)
• (2, 3, 4, 0)
• (3, 4, 0)

Sample Input 2

9
1 5
1 5
1 3
2
3
2
1 6
3
2

Sample Output 2

5
3
5

The i-th line below shows the contents of A after the i-th query is processed in Sample Input 2.

• (5)
• (5, 5)
• (5, 5, 3)
• (5, 3)
• (3, 5)
• (5)
• (5, 6)
• (5, 6)
• (6)

Problem Statement

Find the maximum side length of a regular triangle that can be drawn within a rectangle whose side lengths are A and B.

Constraints

• 1 \leq A,B \leq 1000
• A and B are integers.

Sample Input 1

1 1

Sample Output 1

1.03527618041008295791

The following figure shows an optimal drawing, with the side length of \sqrt{6} - \sqrt{2}.

Note that the sample output does not strictly match \sqrt{6}- \sqrt{2}, but the error is within 10^{-9}, so it is considered correct.