Problem Statement

You are given positive integers A and B.
Let us calculate A+B (in decimal). If it does not involve a carry, print Easy; if it does, print Hard.

Constraints

• A and B are integers.
• 1 \le A,B \le 10^{18}

Sample Input 1

229 390

Sample Output 1

Hard

When calculating 229+390, we have a carry from the tens digit to the hundreds digit, so the answer is Hard.

Sample Input 2

123456789 9876543210

Sample Output 2

Easy

We do not have a carry here; the answer is Easy.
Note that the input may not fit into a 32-bit integer.

Problem Statement

For a positive integer X, let \text{ctz}(X) be the (maximal) number of consecutive zeros at the end of the binary notation of X.
If the binary notation of X ends with a 1, then \text{ctz}(X)=0.

You are given a positive integer N. Print \text{ctz}(N).

Constraints

• 1\leq N\leq 10^9
• N is an integer.

Sample Input 1

2024

Sample Output 1

3

2024 is 11111101000 in binary, with three consecutive 0s from the end, so \text{ctz}(2024)=3.
Thus, print 3.

Sample Input 2

18

Sample Output 2

1

18 is 10010 in binary, so \text{ctz}(18)=1.
Note that we count the trailing zeros.

Sample Input 3

5

Sample Output 3

0

Problem Statement

You observed amoebae and kept some records.

Initially, there was one amoeba, numbered 1.

You made N records. According to the i-th record, the amoeba numbered A_i disappeared by dividing itself into two new amoebae, which were then numbered 2i and 2i+1.
Here, amoeba A_i is said to be the parent of amoebae 2i and 2i+1.

For each k=1,\ldots,2N+1, how many generations away is amoeba k from amoeba 1?

Constraints

• 1 \leq N \leq 2\times 10^5
• The records are consistent. That is:
  • 1\leq A_i \leq 2i-1.
  • A_i are distinct integers.

Sample Input 1

2
1 2

Sample Output 1

0
1
1
2
2

From amoeba 1, amoebae 2 and 3 were born. From amoeba 2, amoebae 4 and 5 were born.

• Amoeba 1 is zero generations away from amoeba 1.
• Amoeba 2 is one generation away from amoeba 1.
• Amoeba 3 is one generation away from amoeba 1.
• Amoeba 4 is one generation away from amoeba 2, and two generations away from amoeba 1.
• Amoeba 5 is one generation away from amoeba 2, and two generations away from amoeba 1.

Sample Input 2

4
1 3 5 2

Sample Output 2

0
1
1
2
2
3
3
2
2

Problem Statement

You are given a positive integer D.

Find the minimum value of |x^2+y^2-D| for non-negative integers x and y.

Constraints

• 1\leq D \leq 2\times 10^{12}
• All input values are integers.

Sample Input 1

21

Sample Output 1

1

For x=4 and y=2, we have |x^2+y^2-D| = |16+4-21|=1.

There are no non-negative integers x and y such that |x^2+y^2-D|=0, so the answer is 1.

Sample Input 2

998244353

Sample Output 2

0

Sample Input 3

264428617

Sample Output 3

32

Problem Statement

You are given positive integers A and B.

You will repeat the following operation until A=B:

• compare A and B to perform one of the following two:
  • if A > B, replace A with A-B;
  • if A < B, replace B with B-A.

How many times will you repeat it until A=B? It is guaranteed that a finite repetition makes A=B.

Constraints

• 1 \le A,B \le 10^{18}
• All values in the input are integers.

Sample Input 1

3 8

Sample Output 1

4

Initially, A=3 and B=8. You repeat the operation as follows:

• A<B, so replace B with B-A=5, making A=3 and B=5.
• A<B, so replace B with B-A=2, making A=3 and B=2.
• A>B, so replace A with A-B=1, making A=1 and B=2.
• A<B, so replace B with B-A=1, making A=1 and B=1.

Thus, you repeat it four times.

Sample Input 2

1234567890 1234567890

Sample Output 2

0

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

Sample Input 3

1597 987

Sample Output 3

15