Problem:
Score : 400 points

Problem Statement

There is a knight - the chess piece - at the origin (0, 0) of a two-dimensional grid.

When the knight is at the square (i, j), it can be moved to either (i+1,j+2) or (i+2, j+1).

In how many ways can the knight reach the square (X, Y)?

Find the number of ways modulo 10^9 + 7.

Constraints

1 \leq X \leq 10^6

1 \leq Y \leq 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

X Y

Output

Print the number of ways for the knight to reach (X, Y) from (0, 0), modulo 10^9 + 7.

Sample Input 1

3 3

Sample Output 1

2

There are two ways: (0,0) \to (1,2) \to (3,3) and (0,0) \to (2,1) \to (3,3).

Sample Input 2

2 2

Sample Output 2

0

The knight cannot reach (2,2).

Sample Input 3

999999 999999

Sample Output 3

151840682

Print the number of ways modulo 10^9 + 7.

Solution in PyPy3 (7.3.0) (judge verdict: Accepted)

def factorial(n, mod):
    fac = [0] * (n + 1)
    inv = [0] * (n + 1)
    fac[0], inv[0] = 1, 1
    for i in range(1, n + 1):
        fac[i] = fac[i-1] * i % mod
        inv[i] = inverse(fac[i], mod)
    return fac, inv

def inverse(a, mod):
    a %= mod # 除数が正なら正になる
    p = mod
    x, y = 0, 1
    while a > 0:
        n = p // a
        p, a = a, p % a, 
        x, y = y, x - n * y
    return x % mod # 除数が正なら正になる

mod = 10 ** 9 + 7
x, y = map(int, input().split())
if (x + y) % 3 != 0:
    print(0)
    quit()
n = (x + y) // 3
if x < n or y < n:
    print(0)
    quit()

m = min(x, y) - n
fac, inv = factorial(n, mod)
print(fac[n] * inv[n-m] * inv[m] % mod) # nCm