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 Rust (1.42.0) (judge verdict: Accepted)

#[allow(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use num::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::collections::*;

#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct ModInt(u32);

impl ModInt {
    pub const MOD: u32 = 1_000_000_007;

    pub fn inv(self) -> Self {
        if self.0 == 0 {
            panic!();
        }
        self.pow(Self::MOD - 2)
    }

    pub fn one() -> Self {
        Self(1)
    }

    pub fn pow(self, e: u32) -> Self {
        if e == 0 {
            return Self::new(1);
        }
        let mut res = self.pow(e >> 1);
        res *= res;
        if e & 1 == 1 {
            res *= self;
        }
        res
    }

    pub fn zero() -> Self {
        Self(0)
    }

    fn new(n: i64) -> Self {
        let mut n = n % (Self::MOD as i64);
        if n.is_negative() {
            n += Self::MOD as i64;
        }
        Self(n as u32)
    }
}

impl From<i32> for ModInt {
    fn from(n: i32) -> Self {
        ModInt::from(n as i64)
    }
}

impl From<i64> for ModInt {
    fn from(n: i64) -> Self {
        Self::new(n)
    }
}

impl From<isize> for ModInt {
    fn from(n: isize) -> Self {
        ModInt::from(n as i64)
    }
}

impl From<u32> for ModInt {
    fn from(n: u32) -> Self {
        ModInt::from(n as u64)
    }
}

impl From<u64> for ModInt {
    fn from(n: u64) -> Self {
        Self::new(n as i64)
    }
}

impl From<usize> for ModInt {
    fn from(n: usize) -> Self {
        ModInt::from(n as u64)
    }
}

impl Into<i32> for ModInt {
    fn into(self) -> i32 {
        self.0 as i32
    }
}

impl Into<i64> for ModInt {
    fn into(self) -> i64 {
        self.0 as i64
    }
}

impl Into<isize> for ModInt {
    fn into(self) -> isize {
        self.0 as isize
    }
}

impl Into<u32> for ModInt {
    fn into(self) -> u32 {
        self.0
    }
}

impl Into<u64> for ModInt {
    fn into(self) -> u64 {
        self.0 as u64
    }
}

impl Into<usize> for ModInt {
    fn into(self) -> usize {
        self.0 as usize
    }
}

impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl std::ops::Add for ModInt {
    type Output = Self;

    fn add(self, rhs: Self) -> Self {
        Self::new((self.0 + rhs.0) as i64)
    }
}

impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl std::ops::Div for ModInt {
    type Output = Self;

    fn div(self, rhs: Self) -> Self {
        self * rhs.inv()
    }
}

impl std::ops::DivAssign for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}

impl std::ops::Mul for ModInt {
    type Output = Self;

    fn mul(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) * (rhs.0 as i64))
    }
}

impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl std::ops::Sub for ModInt {
    type Output = Self;

    fn sub(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) - (rhs.0 as i64))
    }
}

impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

pub struct ModComb {
    max: usize,
    fac: Vec<ModInt>,
    finv: Vec<ModInt>,
}

impl ModComb {
    pub fn new(max: usize) -> Self {
        let mut fac = vec![ModInt::zero(); max + 1];
        let mut finv = vec![ModInt::zero(); max + 1];
        let mut inv = vec![ModInt::zero(); max + 1];
        fac[0] = ModInt::one();
        fac[1] = ModInt::one();
        finv[0] = ModInt::one();
        finv[1] = ModInt::one();
        inv[1] = ModInt::one();
        let modulo = ModInt::MOD as usize;
        for i in 2..=max {
            fac[i] = fac[i - 1] * ModInt::from(i);
            inv[i] =
                ModInt::from(ModInt::from(modulo) - (inv[modulo % i] * ModInt::from(modulo / i)));
            finv[i] = finv[i - 1] * inv[i]
        }
        Self { max, fac, finv }
    }

    pub fn comb(&self, n: usize, k: usize) -> ModInt {
        if n > self.max {
            panic!();
        }
        if n < k {
            return ModInt::zero();
        }
        self.fac[n] * self.finv[k] * self.finv[n - k]
    }
}

fn solve() {
    input! {
        x: usize, y: usize
    };

    let sum = x + y;
    if sum % 3 != 0 {
        println!("0");
        return;
    }
    let all_count = sum / 3;
    let min = x.min(y);
    let max = x.max(y);
    let diff_count = max - min;
    if diff_count > all_count {
        println!("0");
        return;
    }
    let a_count = (all_count - diff_count) / 2;
    let comb = ModComb::new(all_count);
    let res = comb.comb(all_count, a_count);
    println!("{}", res);
}

fn main() {
    std::thread::Builder::new()
        .name("big stack size".into())
        .stack_size(256 * 1024 * 1024)
        .spawn(|| {
            solve();
        })
        .unwrap()
        .join()
        .unwrap();
}