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 Rust (1.15.1) (judge verdict: Time Limit Exceeded)

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        let s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};

    ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($iter, $t);
        input_inner!{$iter $($r)*}
    };
}

macro_rules! read_value {
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };

    ($iter:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
    };

    ($iter:expr, chars) => {
        read_value!($iter, String).chars().collect::<Vec<char>>()
    };

    ($iter:expr, usize1) => {
        read_value!($iter, usize) - 1
    };

    ($iter:expr, $t:ty) => {
        $iter.next().unwrap().parse::<$t>().expect("Parse error")
    };
}

fn main() {
    input! {
        n: usize,
        m: (u32, u32),
        items: [((u32, u32), u32); n],
    }

    let mut costs = items.iter().map(|&((_, _), c)| c).collect::<Vec<_>>();
    costs.sort();

    let mut min = None;

    for i in 1..n + 1 {
        if i >= costs.len() {
            break;
        }
        // println!("--- {:?}", i);
        if let Some(x) = Combinations::new((0..n).collect(), i)
            .flat_map(|group| {
                let ((a, b), c) = group.iter().fold(((0, 0), 0), |acc, i| {
                    let ((i_a, i_b), i_c) = items[*i];
                    let ((a_a, a_b), a_c) = acc;
                    ((i_a + a_a, i_b + a_b), i_c + a_c)
                });
                let g = std::cmp::max(gcd(a, b), 1);
                if (a / g, b / g) == m {
                    Some(c)
                } else {
                    None
                }
            })
            .min()
        {
            // println!("found {:?}", x);
            match min {
                Some(before) => {
                    // println!("before {:?}", before);
                    // println!("lower {:?}", costs.iter().take(i+1).sum::<u32>());
                    if x < before {
                       min = Some(x)
                    }
                    if min.unwrap() <= costs.iter().take(i+1).sum() {
                        // println!("break");
                        break;
                    }
                }
                None => min = Some(x),
            }
        }
    }

    match min {
        Some(x) => println!("{:?}", x),
        None => println!("-1"),
    }
}

fn gcd(a: u32, b: u32) -> u32 {
    match a {
        0 => b,
        _ => gcd(b % a, a),
    }
}

pub struct Combinations<T>
where
    T: Ord + Clone,
{
    original: Vec<T>,
    possition: Vec<usize>,
    len: usize,
    started: bool,
}

impl<T> Combinations<T>
where
    T: Ord + Clone,
{
    pub fn new(mut original: Vec<T>, len: usize) -> Self {
        if original.len() > len && len >= 1 {
            original.sort();
            Combinations {
                original: original,
                possition: (0..len).collect(),
                len: len,
                started: false,
            }
        } else {
            panic!("the length has to be smaller then the datasets len");
        }
    }

    #[inline]
    fn insert(&self, col: &mut Vec<T>) {
        col.clear();
        for (p, n) in self.possition.iter().enumerate() {
            col.insert(p, self.original[*n].clone())
        }
    }

    pub fn next_combination(&mut self, mut comb: &mut Vec<T>) -> bool {
        if !self.started {
            // first pass throught
            self.started = true;
            self.insert(&mut comb);
            true
        } else {
            let org_len = self.original.len();
            // check if we cant bump the back number
            if self.original[self.possition[self.len - 1]] == self.original[org_len - 1] {
                // locate the number closest behind that needs to be bumped
                for i in 2..(self.len + 1) {
                    if self.original[self.possition[self.len - i]] < self.original[org_len - i] {
                        //find the value of the
                        let lastpos = self.possition[self.len - i];
                        let val = &self.original[lastpos];
                        for j in lastpos + 1..org_len {
                            if *val < self.original[j] {
                                for k in 0..i {
                                    self.possition[self.len - i + k] = j + k;
                                }
                                self.insert(&mut comb);
                                return true;
                            }
                        }
                    }
                }
                false
            } else {
                let mut i = self.possition[self.len - 1];
                let current = &self.original[i];
                let mut next = current;
                while current == next {
                    i += 1;
                    next = &self.original[i];
                }
                self.possition[self.len - 1] = i;
                self.insert(&mut comb);
                true
            }
        }
    }
}

impl<T> Iterator for Combinations<T>
where
    T: Ord + Clone,
{
    type Item = Vec<T>;

    fn next(&mut self) -> Option<Self::Item> {
        let mut vals = Vec::with_capacity(self.len);
        if self.next_combination(&mut vals) {
            Some(vals)
        } else {
            None
        }
    }
}