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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using AtCoderBeginnerContest145.Questions;
using AtCoderBeginnerContest145.Extensions;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace AtCoderBeginnerContest145.Questions
{
    public class QuestionD : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var xy = inputStream.ReadIntArray();
            var x = xy[0];
            var y = xy[1];

            if (!IsReachable(x, y))
            {
                yield return 0;
                yield break;
            }

            var totalJumpCount = (x + y) / 3;
            var rightJumpCount = x - totalJumpCount;

            yield return Modular.Combination(totalJumpCount, rightJumpCount).Value;
        }

        bool IsReachable(int x, int y) => x >= y / 2 && y >= x / 2 && (x + y) % 3 == 0;
    }

    [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.LayoutKind.Auto)]
    public struct Modular : IEquatable<Modular>, IComparable<Modular>
    {
        private const int _defaultMod = 1000000007;
        private readonly long _value;    // 0 <= value < Mod の値をとる
        public int Mod { get; }
        public int Value => (int)_value;

        public Modular(long value, int mod = _defaultMod)
        {
            if (mod < 2)
            {
                throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
            }
            Mod = mod;

            if (value >= 0 && value < mod)
            {
                _value = value;
            }
            else
            {
                _value = value % mod;
                if (Value < 0)
                {
                    _value += mod;
                }
            }
        }

        private static void CheckModEquals(Modular a, Modular b)
        {
            if (a.Mod != b.Mod)
            {
                throw new ArgumentException($"{nameof(a)}, {nameof(b)}", $"両者の法（{nameof(Mod)}）は等しくなければなりません。");
            }
        }

        public static Modular operator +(Modular a, Modular b)
        {
            CheckModEquals(a, b);

            var result = a._value + b._value;
            if (result > a.Mod)
            {
                result -= a.Mod;    // 剰余演算を避ける
            }
            return new Modular(result, a.Mod);
        }

        public static Modular operator -(Modular a, Modular b)
        {
            CheckModEquals(a, b);

            var result = a._value - b._value;
            if (result < 0)
            {
                result += a.Mod;    // 剰余演算を避ける
            }
            return new Modular(result, a.Mod);
        }

        public static Modular operator *(Modular a, Modular b)
        {
            CheckModEquals(a, b);
            return new Modular(a._value * b._value, a.Mod);
        }

        public static Modular operator /(Modular a, Modular b)  // こいつは値渡しの方が速いっぽい
        {
            CheckModEquals(a, b);
            return a * Pow(b, a.Mod - 2);
        }

        // 需要は不明だけど一応
        public static bool operator ==(Modular left, Modular right) => left.Equals(right);
        public static bool operator !=(Modular left, Modular right) => !(left == right);
        public static bool operator <(Modular left, Modular right) => left.CompareTo(right) < 0;
        public static bool operator <=(Modular left, Modular right) => left.CompareTo(right) <= 0;
        public static bool operator >(Modular left, Modular right) => left.CompareTo(right) > 0;
        public static bool operator >=(Modular left, Modular right) => left.CompareTo(right) >= 0;

        public static explicit operator int(Modular a) => a.Value;
        public static explicit operator long(Modular a) => a._value;

        public static Modular Pow(Modular a, int n)
        {
            switch (n)
            {
                case 0:
                    return new Modular(1, a.Mod);
                case 1:
                    return a;
                default:
                    var p = Pow(a, n / 2);
                    return p * p * Pow(a, n % 2);
            }
        }

        private static Dictionary<int, List<int>> FactorialCache { get; } = new Dictionary<int, List<int>>();
        private static Dictionary<int, List<int>> PermutationCache { get; } = new Dictionary<int, List<int>>();

        public static Modular Factorial(int n, int mod = _defaultMod)
        {
            if (n < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません。");
            }
            if (mod < 2)
            {
                throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
            }

            if (!FactorialCache.ContainsKey(mod))
            {
                FactorialCache.Add(mod, new List<int>() { 1 });
            }

            var cache = FactorialCache[mod];
            for (int i = cache.Count; i <= n; i++)  // Countが1（0!までキャッシュ済み）のとき1!～n!まで計算
            {
                cache.Add((int)((long)cache[i - 1] * i % mod));
            }
            return new Modular(cache[n], mod);
        }

        public static Modular Combination(int n, int r, int mod = _defaultMod)
        {
            CheckNR(n, r);
            return n == r ? new Modular(1, mod) : Factorial(n, mod) / (Factorial(r, mod) * Factorial(n - r, mod));   // nCr = n! / (n-r)!r! = nPr / n!
        }

        private static void CheckNR(int n, int r)
        {
            if (n <= 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は正の整数でなければなりません。");
            }
            if (r < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(r), $"{nameof(r)}は0以上の整数でなければなりません。");
            }
            if (n < r)
            {
                throw new ArgumentOutOfRangeException($"{nameof(n)},{nameof(r)}", $"{nameof(r)}は{nameof(n)}以下でなければなりません。");
            }
        }

        public override string ToString() => $"{_value} (mod {Mod})";

        public bool Equals(Modular other) => _value == other._value && Mod == other.Mod;

        public override bool Equals(object obj) => obj is Modular && Equals((Modular)obj);

        public int CompareTo(Modular other) => Mod == other.Mod ? _value.CompareTo(other._value) : 0;

        public override int GetHashCode() => Value ^ Mod;
    }

}


namespace AtCoderBeginnerContest145
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionD();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace AtCoderBeginnerContest145.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }

}

#endregion

#region Extensions

namespace AtCoderBeginnerContest145.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion