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 (Python):
X, Y = map(int, input().split())
a = 0
b = 0
for i in range(X + 1):
    N = i
    if (X - N) % 2 != 0 and N != X:
        continue
    M = int((X - N) / 2)
    
    x = N + 2 * M
    y = 2 * N + M
    if X == x and Y == y:
        a = N
        b = M


def cmb(n, r):
    if n - r < r: r = n - r
    if r == 0: return 1
    if r == 1: return n

    numerator = [n - r + k + 1 for k in range(r)]
    denominator = [k + 1 for k in range(r)]

    for p in range(2,r+1):
        pivot = denominator[p - 1]
        if pivot > 1:
            offset = (n - r) % p
            for k in range(p-1,r,p):
                numerator[k - offset] /= pivot
                denominator[k] /= pivot

    result = 1
    for k in range(r):
        if numerator[k] > 1:
            result *= int(numerator[k])

    return result

if a != 0 and b != 0:
    
    ans = cmb(a + b,a) % (10**9+7)
    print(ans)
elif a == 0 and b != 0:
    print("1")
elif a != 0 and b == 0:
    print("1") 
else:
    print("0")

Solution (Python):
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 
N = 10**6
g1 = [1, 1] 
g2 = [1, 1] 
inverse = [0, 1] 

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )
x,y=map(int,input().split())
if x>2*y or y>2*x or (2*x-y)%3!=0 or (2*y-x)%3!=0:
    print(0)
    exit()
print(cmb((x+y)//3,(2*x-y)//3,mod))

Solution (Python):
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 
N = 10**6
g1 = [1, 1] 
g2 = [1, 1] 
inverse = [0, 1] 

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )
x,y=map(int,input().split())
if x>2*y or y>2*x or (2*x-y)%3!=0 or (2*y-x)%3!=0:
    print(0)
    exit()
print(cmb((x+y)//3,(2*x-y)//3,mod))

Solution (Python):
import numpy as np
def cumprod(arr,MOD):
    L = len(arr); Lsq = int(L**.5+1)
    arr = np.resize(arr,Lsq**2).reshape(Lsq,Lsq)
    for n in range(1,Lsq):
        arr[:,n] *= arr[:,n-1]; arr[:,n] %= MOD
    for n in range(1,Lsq):
        arr[n] *= arr[n-1,-1]; arr[n] %= MOD
    return arr.ravel()[:L]

def make_fact(U,MOD):
    x = np.arange(U,dtype=np.int64); x[0] = 1
    fact = cumprod(x,MOD)
    x = np.arange(U,0,-1,dtype=np.int64); x[0] = pow(int(fact[-1]),MOD-2,MOD)
    fact_inv = cumprod(x,MOD)[::-1]
    return fact,fact_inv

U = 10**6
MOD = 10**9 + 7
fact, invfact = make_fact(U,MOD)

X, Y = [int(x) for x in input().split()]
if (X+Y) % 3 != 0:
    print(0)
    exit()
r = (X+Y) // 3
k = X - r
if (k < 0 or Y - r < 0):
    print(0)
else:
    print(fact[r] * invfact[r-k] % MOD * invfact[k] % MOD)

Solution (Python):
def cmb(n,r,mod=10**9+7):
    fac=[1,1]
    for i in range(2,n+1):
        fac.append(fac[i-1]*i%mod)
    ret=fac[n]*pow(fac[r]*fac[n-r]%mod,mod-2,mod)%mod
    return ret

mod = 10**9+7
X, Y = map(int, input().split())
Z = (X+Y) // 3

if (X + Y) % 3 != 0 or abs(X-Y) > Z:
    print(0)
else:
    print(cmb(Z, X-Z))

Solution (Python):

def acinput():
    return list(map(int, input().split(" ")))


def cmb(n, r):
    if n - r < r: r = n - r
    if r == 0: return 1
    if r == 1: return n

    numerator = [n - r + k + 1 for k in range(r)]
    denominator = [k + 1 for k in range(r)]

    for p in range(2,r+1):
        pivot = denominator[p - 1]
        if pivot > 1:
            offset = (n - r) % p
            for k in range(p-1,r,p):
                numerator[k - offset] /= pivot
                denominator[k] /= pivot

    result = 1
    for k in range(r):
        if numerator[k] > 1:
            result *= int(numerator[k])
            
            result%=(10**9+7)

    return result


X,Y=acinput()

import numpy as np
from numpy.linalg import solve
 

x=solve([[1,2],[2,1]],[X,Y])



if np.all(x%1.0==0) and np.all(x>=0):
    print(cmb(int(np.sum(x)),int(np.min(x))))
    

else:
    print(0)

Solution (Python):
def cmb(n, r):
    if n - r < r: r = n - r
    if r == 0: return 1
    if r == 1: return n

    numerator = [n - r + k + 1 for k in range(r)]
    denominator = [k + 1 for k in range(r)]

    for p in range(2,r+1):
        pivot = denominator[p - 1]
        if pivot > 1:
            offset = (n - r) % p
            for k in range(p-1,r,p):
                numerator[k - offset] /= pivot
                denominator[k] /= pivot

    result = 1
    for k in range(r):
        if numerator[k] > 1:
            result *= int(numerator[k])

    return result


x,y=map(int,input().split())
s=int((2*y-x)/3)
t=int((2*x-y)/3)
if (2*y-x)%3!=0 or (2*x-y)%3!=0:
    print(0)
elif s<0 or t<0:
    print(0)
else:
    a = cmb(s+t, t)
    print(a%(10**9+7))

Solution (Python):
import sys

stdin = sys.stdin

ns = lambda : stdin.readline().rstrip()
ni = lambda : int(ns())
na = lambda : list(map(int, stdin.readline().split()))
sys.setrecursionlimit(10 ** 7)

def mod_combination(n, k, mod = 10 ** 9 + 7):
    def extended_gcd(a, b):
        if b == 0:
            return (a, 1, 0)
        d, x, y = extended_gcd(b, a % b)
        return (d, y, x - (a // b) * y)
    p, q = 1, 1
    for i in range(n - k + 1, n + 1):
        p = (p * i) % mod
    for i in range(2, k + 1):
        q = (q * i) % mod
    return p * (extended_gcd(q, mod)[1] % mod) % mod

def main():
  x, y = na()
  if (x + y) % 3 != 0 or x * 2 < y or y * 2 < x:
    print(0)
    return
  n = (x + y) // 3
  print(mod_combination(n, y - n))
 

if __name__ == '__main__':
  main()

Solution (Python):
class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         
        self.inverse = {i: None for i in range(1, N+1)}   
        self.fact_inverse = {i: None for i in range(N+1)} 
        
        
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def perm(self, n, r):
        '''
        Calculate nPr = n! / (n-r)! % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return (self.fact[n] * self.fact_inverse[n-r]) % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod
        
    def hom(self, n, r):
        '''
        Calculate nHr = {n+r-1}Cr % mod.
        Assign r objects to one of n classes.
        Arrangement of r circles and n-1 partitions:
            o o o | o o | | | o | | | o o | | o
        '''
        if n == 0 and r > 0:
            return 0
        if n >= 0 and r == 0:
            return 1
        return self.binom(n + r - 1, r)

MOD = 10**9 + 7
X, Y = map(int, input().split())
if (X + Y) % 3 != 0: print(0)
else:
    x, y = ((2*X - Y) // 3, (-X + 2*Y) // 3)
    com = Combinatorics(x + y, MOD)
    print(com.binom(x+y, x))

Solution (Python):
def main():
    mod=1000000007
    x,y=map(int,input().split())
    if (x+y)%3:
        return 0
    
    n=(x+y)//3
    Fact=[1]
    for i in range(1,n+1):
        Fact.append(Fact[i-1]*i%mod)
    Finv=[0]*(n+1)
    Finv[-1]=pow(Fact[-1],mod-2,mod)
    for i in range(n-1,-1,-1):
        Finv[i]=Finv[i+1]*(i+1)%mod
    def comb(n,r):
        if n<r:
            return 0
        return Fact[n]*Finv[r]*Finv[n-r]%mod
    
    a=(-x+2*y)//3
    b=(x*2-y)//3
    if a<0 or b<0:
        return 0
    return comb(a+b,a)
    
if __name__=='__main__':
    print(main())

Solution (Python):
import math
from scipy.misc import comb

def framod(n, mod, a=1):
    for i in range(1,n+1):
        a=a * i % mod
    return a

def power(n, r, mod):
    if r == 0: return 1
    if r%2 == 0:
        return power(n*n % mod, r//2, mod) % mod
    if r%2 == 1:
        return n * power(n, r-1, mod) % mod

def comb(n, k, mod):
    a=framod(n, mod)
    b=framod(k, mod)
    c=framod(n-k, mod)
    return (a * power(b, mod-2, mod) * power(c, mod-2, mod)) % mod

x, y = (int(i) for i in input().split())
af, a = math.modf((2*y-x)/3)
bf, b = math.modf((2*x-y)/3)
a, b = int(a), int(b)
if (af != 0) or (bf != 0) or (a < 0) or (b < 0):
    print(0)
else:
    aa, bb = min(a, b), max(a, b)
    print(comb(aa+bb, aa, 10**9+7))

Solution (Python):
X,Y = map(int,input().split())

MOD = 10**9 + 7
FAC = [1]
INV = [1]
for i in range(1,X+Y+1):
  FAC.append((FAC[i-1]*i) % MOD)
  INV.append(pow(FAC[-1],MOD-2,MOD))

def nCr(n,r):
  return FAC[n]*INV[n-r]*INV[r]

a = (2*X-Y)/3
b = (2*Y-X)/3
if a>=0 and b>=0 and a.is_integer() and b.is_integer():
  print(nCr(int(a)+int(b),min(int(a),int(b)))%MOD)
else:
  print(0)

Solution (Python):


X,Y=map(int, raw_input().split())

mod=10**9+7
fact, inv_fact = [1], [1]
fact_tmp = 1
for i in range(1, max(X,Y)+1):
    fact_tmp *= i
    fact_tmp %= mod
    fact.append(fact_tmp)
    inv_fact.append(pow(fact_tmp, mod-2, mod))
 
 
def ncr(n,r):
	if n < 0 or r < 0 or n < r:	return 0
	else:	return (fact[n] * inv_fact[r] * inv_fact[n-r]) %mod

if (X+Y)%3!=0:
    print 0
else:
    m=(2*X-Y)/3
    n=(Y-m)/2
    print ncr(m+n,n)

Solution (Python):
import math
import sys


def Knight():
    def check(d):
        if (d[0] + d[1]) % 3 or d[0] < 0 or d[1] < 0:
            print(0)
            sys.exit()
    d = list(map(int, input().split()))
    check(d)

    x = d[0]
    y = d[1]
    diff = x - y if x > y else y - x  
    d2 = [x - diff, y - diff * 2] if x < y else [x - diff * 2, y - diff]
    r = d2[0] // 3

    def cmb(n, r, mod):
        if (r < 0 or r > n):
            return 0
        r = min(r, n-r)
        return g1[n] * g2[r] * g2[n-r] % mod

    mod = 10**9+7  
    N = x if x > y else y
    g1 = [1, 1]  
    g2 = [1, 1]  
    inverse = [0, 1]  

    for i in range(2, N + 1):
        g1.append((g1[-1] * i) % mod)
        inverse.append((-inverse[mod % i] * (mod//i)) % mod)
        g2.append((g2[-1] * inverse[-1]) % mod)

    a = cmb(r*2+diff, r, mod)

    print(a)


Knight()

Solution (Python):
class Combination:
    def __init__(self, size, mod=10**9 + 7):
        self.size = size + 2
        self.mod = mod
        self.fact = [1, 1] + [0] * size
        self.factInv = [1, 1] + [0] * size
        self.inv = [0, 1] + [0] * size

        for i in range(2, self.size):
            self.fact[i] = self.fact[i - 1] * i % self.mod
            self.inv[i] = -self.inv[self.mod % i] * (self.mod // i) % self.mod
            self.factInv[i] = self.factInv[i - 1] * self.inv[i] % self.mod

    def npr(self, n, r):
        if n < r or n < 0 or r < 0:
            return 0
        return self.fact[n] * self.factInv[n - r] % self.mod

    def ncr(self, n, r):
        if n < r or n < 0 or r < 0:
            return 0
        return self.fact[n] * (self.factInv[r] * self.factInv[n - r] % self.mod) % self.mod

    def factN(self, n):
        if n < 0:
            return 0
        return self.fact[n]

    def f(self, p, q):
        return self.fact[p + q] * self.factInv[p] * self.factInv[q] % MOD


X, Y = map(int, input().split())
MOD = 10**9 + 7

if (X + Y) % 3 != 0:
    print(0)
    exit()

time = (X + Y) // 3
ans = 0
comb = Combination(time + 10)

for y in range(time + 1):
    x = time - y

    if X == x * 2 + y and Y == y * 2 + x:
        ans += comb.f(x, y)
        ans %= MOD

print(ans % MOD)

Solution (Python):
import sys
input = sys.stdin.readline


X, Y = map(int, input().split())


import math

def cmb(n, r):
    if n - r < r: r = n - r
    if r == 0: return 1
    if r == 1: return n

    numerator = [n - r + k + 1 for k in range(r)]
    denominator = [k + 1 for k in range(r)]

    for p in range(2,r+1):
        pivot = denominator[p - 1]
        if pivot > 1:
            offset = (n - r) % p
            for k in range(p-1,r,p):
                numerator[k - offset] /= pivot
                denominator[k] /= pivot

    result = 1
    for k in range(r):
        if numerator[k] > 1:
            result *= int(numerator[k])

    return result

if (X + Y) % 3 == 0:
    k = (-X + 2*Y) //3
    l = (2*X -Y) //3
    if k < 0 or l <0:
        answer = 0
    else:
        answer = cmb(k+l, l) % (10**9 + 7)
else:
    answer = 0

print(answer)

Solution (Python):
X,Y=map(int,input().split())
import math

def combination(n, r, mod=10**9+7):
    n1, r = n+1, min(r, n-r)
    numer = denom = 1
    for i in range(1, r+1):
        numer = numer * (n1-i) % mod
        denom = denom * i % mod
    return numer * pow(denom, mod-2, mod) % mod

def f2(r):

    return inv_t
def co(n,r):
    P = 10**9 + 7
    N = r+1
    inv_t = [0]+[1]
    for i in range(2,N+1):
        inv_t += [inv_t[P % i] * (P - int(P / i)) % P]
    ans =1
    for i in range(1,r+1):
        ans = (ans * (n+i-1)) % P
    for i in range(1,r+1):
        ans = (ans * inv_t[i]) % P
    return ans
if (X+Y)%3==0:
    Z=X+Y
    if min(X,Y)-Z//3 >-1:
        
        print(combination(Z//3,min(X,Y)-Z//3))
        
    else:
        print(0)
else:
    print(0)

Solution (Python):
import math
X, Y = map(int, input().split())
p = (((X+Y)/3)+Y-X)/2
m = p+X-Y


def framod(n, mod, a=1):
    for i in range(1,n+1):
        a=a * i % mod
    return a

def power(n, r, mod):
    if r == 0: return 1
    if r%2 == 0:
        return power(n*n % mod, r//2, mod) % mod
    if r%2 == 1:
        return n * power(n, r-1, mod) % mod

def comb(n, k, mod):
    a=framod(n, mod)
    b=framod(k, mod)
    c=framod(n-k, mod)
    return (a * power(b, mod-2, mod) * power(c, mod-2, mod)) % mod


if p % 1 != 0 or p<0 or m<0:
    ans = 0

else:
    q = p+m
    q = int(q)
    p = int(p)
    ans = comb(q,p,10**9+7)
  

print(ans)

Solution (Python):

import sys,itertools
def input():
    return sys.stdin.readline()[:-1]

X,Y = map(int,input().split(' '))
u = 10**9+7

S = X+Y

class Combination(object):
        """参考: https://harigami.net/contents?id=5f169f85-5707-4137-87a5-f0068749d9bb"""
        __slots__ = ['mod', 'factorial', 'inverse']

        def __init__(self, max_val_arg: int = 10**6, mod: int = 10**9 + 7):
            fac, inv = [1], []
            fac_append, inv_append = fac.append, inv.append

            for i in range(1, max_val_arg + 1):
                fac_append(fac[-1] * i % mod)

            inv_append(pow(fac[-1], mod - 2, mod))

            for i in range(max_val_arg, 0, -1):
                inv_append((inv[-1] * i) % mod)

            self.mod, self.factorial, self.inverse = mod, fac, inv[::-1]

        def combination(self, n, r):
            if n < 0 or r < 0 or n < r:
                return 0
            return self.factorial[n] * self.inverse[r] * self.inverse[n - r] % self.mod

if not S%3==0:
    print(0)
    exit()
else:
    small = S//3
    s_elm = min(X,Y)
    if s_elm<small:
        print(0)
        exit()
    r = s_elm - small
    ans = Combination().combination(small, r)
    ans %= u
print(ans)

Solution (Python):
class Solution:
    def solve(self, x: int, y: int) -> int:

        if (2*x - y) % 3 != 0 or (-x + 2*y) % 3 != 0:
            return 0

        m = (2*x - y) // 3
        n = (-x + 2*y) // 3

        if m < 0 or n < 0:
            return 0

        
        def egcd(a, b):
            if a == 0:
                return b, 0, 1
            else:
                g, y, x = egcd(b % a, a)
                return g, x - (b // a) * y, y

        def modinv(a, m):
            g, x, y = egcd(a, m)
            if g != 1:
                raise Exception('modular inverse does not exist')
            else:
                return x % m

        def convination(n: int, r: int, mod: int = 10**9+7) -> int:
            r = min(r, n-r)
            res = 1
            for i in range(r):
                res = res * (n-i) * modinv(i+1, mod) % mod
            return res

        return convination(n+m, m)


if __name__ == '__main__':

    
    x, y = map(int, input().split())

    
    solution = Solution()
    print(solution.solve(x, y))

Solution (Python):
import math
import sys
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10 ** 9 + 7
N = 10**6
g1 = [1, 1]
g2 = [1, 1]
inverse = [0, 1]
for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

    
x, y = map(int, input().split())
if x  > 2 * y:
    print("0")
    sys.exit()
elif y  > 2 * x:
    print("0")
    sys.exit()
elif (x + y) % 3 != 0:
    print("0")
    sys.exit()


z = x + y
a = int(z / 3)
n = a
b = 2 * a
r = 0
ans = 1
while (True):
    if a == x and b == y:
        break
    a += 1
    b -= 1
    r += 1



print(cmb(n, r, mod))

Solution (Python):
import sys
input = sys.stdin.readline
from operator import itemgetter
sys.setrecursionlimit(10000000)
INF = 10**30
from math import factorial

fac = [0] * (10**6+5)
ifac = [0] * (10**6+5)
d = 10**9 + 7

def mpow(x, n):
    ans = 1
    while n != 0:
        if n&1:
            ans = ans * x % d
        x = x*x % d
        n = n >> 1
    return ans

def comb(a, b):
    if a == 0 and b == 0:
        return 1
    if a < b or a < 0:
        return 0
    tmp = ifac[a-b] * ifac[b] % d
    return tmp * fac[a] % d

def main():
    global fac, ifac
    x, y = map(int, input().strip().split())
    if (x + y) % 3 != 0:
        print(0)
        return
    k = (x+y) // 3
    q = x - k
    if x - 2 * q < 0 or q < 0:
        print(0)
        return

    fac[0] = 1
    ifac[0] = 1
    for i in range(10**6+1):
        fac[i+1] = fac[i]*(i+1) % d
        ifac[i+1] = ifac[i]*mpow(i+1, d-2) % d

    print(comb(k, q) % d)

if __name__ == '__main__':
    main()

Solution (Python):
import sys
read = sys.stdin.buffer.read
input=lambda :sys.stdin.readline().rstrip()
inputs= sys.stdin.buffer.readlines
INF=float("inf")
MOD=10**9+7

sys.setrecursionlimit(2147483647)






















def framod(n, mod, a=1):
	for i in range(1, n + 1):
		a = a * i % mod
	return a


def power(n, r, mod):
	if r == 0: return 1
	if r % 2 == 0:
		return power(n * n % mod, r // 2, mod) % mod
	if r % 2 == 1:
		return n * power(n, r - 1, mod) % mod


def comb(n, k, mod):
	a = framod(n, mod)
	b = framod(k, mod)
	c = framod(n - k, mod)
	return (a * power(b, mod - 2, mod) * power(c, mod - 2, mod)) % mod


def main():
	X,Y=map(int,input().split())
	
	a=max(X,Y)
	b=min(X,Y)
	
	ans=0
	
	t=2*a-b
	
	
	if t%3==0 and t>0 and a<=b*2:
		c1=t//3
		c2=a-2*t//3
		ans=comb(c1+c2,c1,MOD)
		
	print(ans)
		
	
	
	
	
if __name__ == '__main__':
	main()

Solution (Python):
from functools import reduce

def modpow(a, m):
    ret = 1
    while m > 0:
        if m & 1:
            ret = ret * a % mod
        a = a * a % mod
        m = m >> 1
    return ret

def modinv(a):
    return modpow(a, mod - 2)

def cmb(n, r):
    r = min(r, n - r)
    if r == 0:
        return 1
    over = reduce(lambda a, b: a * b % mod, range(n, n - r, -1))
    under = reduce(lambda a, b: a * b % mod, range(1, r + 1))
    return over * modinv(under) % mod

x, y = [int(i) for i in input().split()]
if (x + y) % 3 or x > 2 * y or y > 2 * x:
    print(0)
else:
    mod = 10**9 + 7
    print(cmb((x + y) // 3, (2 * x - y) // 3))

Solution (Python):
import math
mod = (10**9) + 7

def kaijo(n):
    result = 1
    for i in reversed(range(1,n+1)):
        result = (result * i) % mod
    return result
def combinations_count(n, r):
    return (kaijo(n)) //(kaijo(n - r) * kaijo(r))

def nCk(n, k):
    ret = 1
    mo = 10**9 + 7
    for i in range(k):
        ret = ret * (n - i) * pow(i + 1, mo - 2, mo) % mo
    return ret


X,Y = map(int,input().split(' '))

if (X+Y)%3 == 0:
    flag = False
    if X > Y:
        diff = X - Y
        all_num = (X+Y)//3
        count_2 = Y - all_num
        if X > Y*2:
            flag = True
    elif Y > X:
        diff = Y - X
        all_num = (X+Y)//3
        count_2 = X - all_num
        if Y > X*2:
            flag = True
    else:
        all_num = (X+Y)//3
        count_2 = all_num//2
    if flag:
        print(0)
    else:
        print(nCk(all_num,count_2))
else:
    print(0)

Solution (Python):
import math
MOD = 1000000000+7
X, Y = map(int, input().split())
 
if ((2*Y-X)%3 != 0) or ((2*X-Y)%3 != 0) or (2*Y-X) < 0 or (2*X-Y) < 0:
    print(0)
elif 2 *Y == X or 2 * X == Y:
    print(1)
else:
    x = (2*Y-X) // 3
    y = (2*X-Y) // 3
    fac_list = [None for _ in range(x+y+2)]
    inv_fac_list = [None for _ in range(x+y+2)]
 
    fac_list[0] = 1
    for i in range(1, x+y+2):
        fac_list[i] = i * fac_list[i-1] % MOD
 
    inv_fac_list[x+y+1] = pow(fac_list[x+y+1],(MOD - 2), MOD)
    for i in range(x+y, 0, -1):
        inv_fac_list[i] = inv_fac_list[i+1] * (i+1) % MOD
 
    print((fac_list[x+y] * inv_fac_list[x] * inv_fac_list[y]) % MOD)

Solution (Python):
import sys
G = list(map(int, input().split()))
total = G[0] + G[1]
if total % 3 != 0:
  print(0)
  sys.exit(0)

total //= 3
dis = G[0] - total

MOD = 10 ** 9 + 7
def inv_mod(a, p=MOD):
    def inv_mod_sub(a, p):
        if a == 1:
            return 1, 0
        else:
            d, r = p//a, p%a
            x, y = inv_mod_sub(r, a)
            return y-d*x, x
    if p < 0: p = -p
    a %= p
    return inv_mod_sub(a,p)[0] % p

def comb_mod(n, k):
    if k < 0 or k > n:
        return 0
    else:
        return f_mod[n]*f_mod_inverse[k]*f_mod_inverse[n-k] % MOD
f_mod=[1]*(10**6)
f_mod_inverse=[1]*(10**6)
for i in range(1,10**6):
  f_mod[i]=f_mod[i-1]*i%MOD
  f_mod_inverse[i]=f_mod_inverse[i-1]*inv_mod(i)%MOD

print(comb_mod(total, dis))

Solution (Python):
def solve():
    X, Y = map(int, input().split())
    if (2 * Y - X) % 3 > 0 or (2 * X - Y) % 3 > 0: print(0)
    else:
        a = (2 * Y - X) // 3
        b = (2 * X - Y) // 3
        if a >= 0 and b >= 0:
            mod = 7 + 10 ** 9
            fact = [0] * (2 * 10 ** 6 + 1)
            revfact = [0] * (2 * 10 ** 6 + 1)
            fact[1] = 1
            for i in range(2, 2 * 10 ** 6 + 1):
                fact[i] = (i * fact[i-1]) % mod
            revfact[2 * 10 ** 6] = pow(fact[2 * 10 ** 6], mod - 2, mod)
            for i in reversed(range(2 * 10 ** 6)):
                revfact[i] = ((i + 1) * revfact[i + 1]) % mod

            print((fact[a + b] * revfact[a] * revfact[b]) % mod)
        else: print(0)

    

    return 0

if __name__ == "__main__":
    solve()

Solution (Python):
import sys

input = sys.stdin.readline

ri = lambda: int(input())
rs = lambda: input().rstrip()
ril = lambda: list(map(int, input().split()))
rsl = lambda: input().rstrip().split()
ris = lambda n: [ri() for _ in range(n)]
rss = lambda n: [rs() for _ in range(n)]
rils = lambda n: [ril() for _ in range(n)]
rsls = lambda n: [rsl() for _ in range(n)]


class Combinations:
    def __init__(self, max_num, mod):
        self.mod = mod
        self.factorials = [1]
        for i in range(1, max_num + 1):
            self.factorials.append((self.factorials[-1] * i) % mod)
        self.invs = [pow(self.factorials[-1], mod - 2, mod)]
        for i in reversed(range(1, max_num + 1)):
            self.invs.append(self.invs[-1] * i % mod)
        self.invs = self.invs[::-1]

    def __call__(self, n, k):
        if n < 0 or k < 0 or k > n:
            return 0
        return self.factorials[n] * self.invs[k] * self.invs[n - k] % self.mod


MOD = 10**9 + 7

x, y = ril()
a = (2 * x - y) // 3
b = a - x + y
if 3 * a != 2 * x - y:
    print(0)
    exit()

combinations = Combinations(10**6, MOD)
print(combinations(a + b, a))

Solution (Python):
x,y=map(int,input().split())
MOD=10**9+7
ans=0
if y>x:
    x,y=y,x

def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = MOD 
N = 10**6
g1 = [1, 1] 
g2 = [1, 1] 
inverse = [0, 1] 

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

for a in range(1,x//2+1):
    b=x-2*a
    if 2*b+a==y:
        tmp=cmb(b+a,a,mod)%MOD
        ans+=tmp
        ans%=MOD

print(ans)

Solution (Python):
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

x,y=map(int,input().split())

if (2*y-x)%3==0:
    a=(2*y-x)//3
    b=(2*x-y)//3
    mod = 10**9+7 
    N = a+b
    r=min(a,b)
    g1 = [1, 1] 
    g2 = [1, 1] 
    inverse = [0, 1] 

    for i in range( 2, N + 1 ):
        g1.append( ( g1[-1] * i ) % mod )
        inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
        g2.append( (g2[-1] * inverse[-1]) % mod )

    print(cmb(N,r,mod))
else:
    print(0)

Solution (Python):

x,y = map(int, input().split())

from operator import mul
from functools import reduce

def combinations_count(n, r):
    r = min(r, n - r)
    numer = reduce(mul, range(n, n - r, -1), 1)
    denom = reduce(mul, range(1, r + 1), 1)
    return numer // denom

def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 
N = 10**6
g1 = [1, 1] 
g2 = [1, 1] 
inverse = [0, 1] 

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )



move = (x+y) / 3
if int(move) != move or x < move or y < move:
    print(0)
else:
    move = int(move)
    diff = max(x,y) - min(x,y)
    res = cmb(move, move//2 - diff//2, mod)
    print(res)

Solution (Python):
inf = 10**9+7
import sys
a = 0
def nCr(n, r):
    r = min(r, n - r)
    if r == 0: return 1;
    if r == 1: return n;
    numerator = [n - r + i + 1 for i in range(r)]
    denominator  = [i + 1 for i in range(r)]
    for p in range(2, r + 1):
        pivot = denominator[p - 1]
        if pivot > 1:
            offset = (n - r) % p
            for k in range(p - 1, r, p):
                numerator[k - offset] /= pivot
                denominator[k] /= pivot
    result = 1
    for k in range(r):
        if numerator[k] > 1: 
            result = (result * numerator[k])%inf
    return result

x, y = map(int, input().split())
key = 0
if x == 2*y:
    print(1)
    sys.exit()
if y == 2*x:
    print(1)
    sys.exit()
while x>0 and y>0:
    x, y = x-3, y-3
    key += 1
    if x == 2*y:
        a = 1
        break
    if y == 2*x:
        a = -1
        break

if x < 0 or y < 0:
    print(0)
    sys.exit()

if a == 0:
    print(0)
elif a == 1:
    print(int(nCr(key*2+y ,key)))
else:
    print(int(nCr(key*2+x ,key)))

Solution (Python):
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7
N = 10**6
g1 = [1, 1]
g2 = [1, 1]
inverse = [0, 1]

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )
x,y = map(int,input().split())
a = (x+y)//3
if (x+y)%3:
    print(0)
elif x < a or y < a:
    print(0)
else:
    b = min(x-a,y-a)
    if b:
        print(cmb(a,b,mod))
    else:
        print(1)

Solution (Python):
x,y=map(int,input().split())
a=(2*y-x)/3
b=(2*x-y)/3
mod=10**9+7
if a==int(a) and b==int(b) and a>=0 and b>=0:
  def mul(x,y):
    return ((x%mod)*(y%mod))%mod
  
  def power(x,y):
    if y==0:
      return 1
    elif y==1:
      return x%mod
    elif y%2==0:
      return power(x,y//2)**2%mod
    else:
      return power(x,y//2)**2*x%mod
    
  def div(x,y):
    return mul(x,power(y,mod-2))
  
  p=1
  for i in range(int(a+1),int(a+b+1)):
    p=mul(p,i)
  q=1
  for i in range(1,int(b+1)):
    q=mul(q,i)
  print(div(p,q))
else:
  print(0)

Solution (Python):
X, Y = map(int, input().split())
MOD = 10**9+7
N = 134*10**4

def power(base, num):
    if num == 0:
        return 1
    elif num % 2==0:
        return power(base, num//2) **2 % MOD
    elif num==1:
        return base % MOD
    else:
        return power(base, num//2) **2 * base % MOD
    
fact = [0 for i in range(N)]
inv_fact = [0 for i in range(N)]
 
fact[0] = 1
for i in range(1, N):
    fact[i] = fact[i-1] * i % MOD

inv_fact[N-1] = power(fact[N-1], MOD-2)
for i in range(1, N):
    inv_fact[N-1-i] = inv_fact[N-i] * (N-i) % MOD
 
def comb(x,y):
    return (fact[x] * inv_fact[y])%MOD * inv_fact[x-y] % MOD

if (2*X-Y)%3==0 and (-X+2*Y)%3==0 and 2*X-Y>=0 and -X+2*Y>=0:
    a = (2*X-Y) //3
    b = (-X+2*Y)//3
    ans = comb(a+b, min(a,b)) % MOD

    print(ans)

    
else:
    print(0)

Solution (Python):
def powermod(a, p, mod):
    if p == 0:
        return 1
    elif p % 2 == 0:
        return (powermod(a, p//2, mod) ** 2) % mod
    else:
        return ((powermod(a, (p-1)//2, mod) ** 2) % mod) * a % mod

def combinationmod(n,r):
    if r == 0 or n == r:
        return 1
    mod = 10 ** 9 + 7
    factorial = 1
    for i in range(n):
        factorial = (factorial * (i+1)) % mod
        if i+1 == r:
            fact_1 = factorial
        if i+1 == n-r:
            fact_2 = factorial
        if i+1 == n:
            fact_0 = factorial
    return fact_0 * powermod(fact_1,mod-2,mod) % mod * powermod(fact_2,mod-2,mod) % mod
    
X,Y = map(int,input().split())
if (X + Y) % 3 == 0 and X <= 2 * Y and 2 * X >= Y:
    turns = (X + Y) // 3
    print(combinationmod(turns, min(X,Y) - turns))
else:
    print(0)

Solution (Python):
X,Y=map(int,input().split())

def framod(n, mod, a=1):
    for i in range(1,n+1):
        a=a * i % mod
    return a

def power(n, r, mod):
    if r == 0: return 1
    if r%2 == 0:
        return power(n*n % mod, r//2, mod) % mod
    if r%2 == 1:
        return n * power(n, r-1, mod) % mod

def comb(n, k, mod):
    a=framod(n, mod)
    b=framod(k, mod)
    c=framod(n-k, mod)
    return (a * power(b, mod-2, mod) * power(c, mod-2, mod)) % mod

Z=((X+Y)//3)

if (X+Y)%3==0 and abs(X-Y)<=Z:
  ans=(comb(Z, X-Z,(10**9+7)))
  print(ans)
else:
  print(0)