Problem:
Score : 400 points

Problem Statement

Dolphin is planning to generate a small amount of a certain chemical substance C.

In order to generate the substance C, he must prepare a solution which is a mixture of two substances A and B in the ratio of M_a:M_b.

He does not have any stock of chemicals, however, so he will purchase some chemicals at a local pharmacy.

The pharmacy sells N kinds of chemicals. For each kind of chemical, there is exactly one package of that chemical in stock.

The package of chemical i contains a_i grams of the substance A and b_i grams of the substance B, and is sold for c_i yen (the currency of Japan).

Dolphin will purchase some of these packages. For some reason, he must use all contents of the purchased packages to generate the substance C.

Find the minimum amount of money required to generate the substance C.

If it is not possible to generate the substance C by purchasing any combination of packages at the pharmacy, report that fact.

Constraints

1≦N≦40

1≦a_i,b_i≦10

1≦c_i≦100

1≦M_a,M_b≦10

gcd(M_a,M_b)=1

a_i, b_i, c_i, M_a and M_b are integers.

Input

The input is given from Standard Input in the following format:

N M_a M_b
a_1 b_1 c_1
a_2 b_2 c_2
:
a_N b_N c_N

Output

Print the minimum amount of money required to generate the substance C. If it is not possible to generate the substance C, print -1 instead.

Sample Input 1

3 1 1
1 2 1
2 1 2
3 3 10

Sample Output 1

3

The amount of money spent will be minimized by purchasing the packages of chemicals 1 and 2.

In this case, the mixture of the purchased chemicals will contain 3 grams of the substance A and 3 grams of the substance B, which are in the desired ratio: 3:3=1:1.

The total price of these packages is 3 yen.

Sample Input 2

1 1 10
10 10 10

Sample Output 2

-1

The ratio 1:10 of the two substances A and B cannot be satisfied by purchasing any combination of the packages. Thus, the output should be -1.

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

using System;
using System.Collections.Generic;
using System.Linq;

class Program
{
    static int N;
    static int Ma, Mb;
    static int[] a;
    static int[] b;
    static int[] c;
    public static void Main(string[] args)
    {
        checked
        {
            N = Input.NextInt();
            Ma = Input.NextInt();
            Mb = Input.NextInt();

            a = new int[N];
            b = new int[N];
            c = new int[N];

            for (int i = 0; i < N; i++)
            {
                var s = Input.LineInt();
                a[i] = s[0];
                b[i] = s[1];
                c[i] = s[2];
            }

            var maxa = a.Sum();
            var maxb = b.Sum();

            DPs = new Dictionary<Tuple<int, int, int>, int>();

            int minCost = BigValue;
            for (int i = 1; ; i++)
            {
                var mma = Ma * i;
                var mmb = Mb * i;
                if (mma > maxa)
                    break;
                if (mmb > maxb)
                    break;

                var cost = DP(N, mma, mmb);
                minCost = Math.Min(minCost, cost);
            }

            if (minCost == BigValue)
                minCost = -1;

            Console.WriteLine(minCost);
        }
    }

    public const int BigValue = int.MaxValue;

    static Dictionary<Tuple<int, int, int>, int> DPs;

    static int DP(int i, int ca, int cb)
    {
        var key = new Tuple<int, int, int>(i, ca, cb);
        if (DPs.ContainsKey(key))
            return DPs[key];

        if (i == 0)
        {
            if (ca == 0 && cb == 0)
                return 0;
            else
                return BigValue;
        }

        int cost1;
        {
            var key1 = new Tuple<int, int, int>(i - 1, ca, cb);
            cost1 = DP(i - 1, ca, cb);
        }

        int cost2;
        {
            var mca = ca - a[i - 1];
            var mcb = cb - b[i - 1];
            if (mca < 0 || mcb < 0)
            {
                cost2 = BigValue;
            }
            else
            {
                var key2 = new Tuple<int, int, int>(i - 1, mca, mcb);
                cost2 = DP(i - 1, mca, mcb);
                if (cost2 != BigValue)
                    cost2 += c[i - 1];
            }
        }

        var cost = Math.Min(cost1, cost2);
        DPs[key] = cost;
        return cost;
    }
}

public class MathLib
{
    /// <summary>
    /// 最大公約数を求める
    /// </summary>
    /// <param name="a"></param>
    /// <param name="b"></param>
    /// <returns></returns>
    public static long Gcd(long a, long b)
    {
        return (a < b) ? Gcd(b, a)
             : (b > 0) ? Gcd(b, a % b)
                       : a;
    }

    /// <summary>
    /// 最小公倍数を求める
    /// </summary>
    /// <param name="a"></param>
    /// <param name="b"></param>
    /// <returns></returns>
    public static long Lcm(long a, long b)
    {
        return a / Gcd(a, b) * b;
    }

    /// <summary>
    /// 約数を列挙する。返される順序は一様では<b>ない</b>。
    /// </summary>
    /// <param name="n">約数を求める数字</param>
    /// <returns><paramref name="n"/> の約数の列挙</returns>
    public static IEnumerable<int> GetDivisors(int n)
    {
        for (int i = 1; i < (int)Math.Sqrt(n) + 1; i++)
        {
            if ((n % i) == 0)
            {
                yield return i;
                if (i != n / i)
                {
                    yield return n / i;
                }
            }
        }
        yield break;
    }
}

public static class Input
{
    private static Queue<string> q = new Queue<string>();
    private static void Confirm() { if (q.Count == 0) foreach (var s in Console.ReadLine().Split(' ')) q.Enqueue(s); }
    public static int NextInt() { Confirm(); return int.Parse(q.Dequeue()); }
    public static long NextLong() { Confirm(); return long.Parse(q.Dequeue()); }
    public static string NextString() { Confirm(); return q.Dequeue(); }
    public static double NextDouble() { Confirm(); return double.Parse(q.Dequeue()); }
    public static int[] LineInt() { return Console.ReadLine().Split(' ').Select(int.Parse).ToArray(); }
    public static long[] LineLong() { return Console.ReadLine().Split(' ').Select(long.Parse).ToArray(); }
    public static string[] LineString() { return Console.ReadLine().Split(' ').ToArray(); }
    public static double[] LineDouble() { return Console.ReadLine().Split(' ').Select(double.Parse).ToArray(); }
}