Problem:
Score : 600 points

Problem Statement

Takahashi wants to grill N pieces of meat on a grilling net, which can be seen as a two-dimensional plane. The coordinates of the i-th piece of meat are \left(x_i, y_i\right), and its hardness is c_i.

Takahashi can use one heat source to grill the meat. If he puts the heat source at coordinates \left(X, Y\right), where X and Y are real numbers, the i-th piece of meat will be ready to eat in c_i \times \sqrt{\left(X - x_i\right)^2 + \left(Y-y_i\right)^2} seconds.

Takahashi wants to eat K pieces of meat. Find the time required to have K or more pieces of meat ready if he put the heat source to minimize this time.

Constraints

All values in input are integers.

1 \leq N \leq 60

1 \leq K \leq N

-1000 \leq x_i , y_i \leq 1000

\left(x_i, y_i\right) \neq \left(x_j, y_j\right) \left(i \neq j \right)

1 \leq c_i \leq 100

Input

Input is given from Standard Input in the following format:

N K
x_1 y_1 c_1
\vdots
x_N y_N c_N

Output

Print the answer.

It will be considered correct if its absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

4 3
-1 0 3
0 0 3
1 0 2
1 1 40

Sample Output 1

2.4

If we put the heat source at \left(-0.2, 0\right), the 1-st, 2-nd, and 3-rd pieces of meat will be ready to eat within 2.4 seconds. This is the optimal place to put the heat source.

Sample Input 2

10 5
-879 981 26
890 -406 81
512 859 97
362 -955 25
128 553 17
-885 763 2
449 310 57
-656 -204 11
-270 76 40
184 170 16

Sample Output 2

7411.2252

Solution in Rust (1.15.1) (judge verdict: Accepted)

fn read_stdin() -> String {
    use std::io::Read;
    let mut input = String::new();
    std::io::stdin().read_to_string(&mut input).unwrap();
    input
}

fn scan<'a, I: Iterator<Item = &'a str>, T: std::str::FromStr>(tokens: &mut I) -> T {
    tokens.next().unwrap().parse().ok().unwrap()
}

fn vec<T, F: FnMut(usize) -> T>(n: usize, mut f: F) -> Vec<T> {
    (0..n).map(|i| f(i)).collect()
}

const EPS: f64 = 1e-6;

fn main() {
    let input = read_stdin();
    let tokens = &mut input.split_whitespace();
    let n: usize = scan(tokens);
    let k: usize = scan(tokens);
    let xyc: Vec<(f64, f64, f64)> = vec(n, |_| (scan(tokens), scan(tokens), scan(tokens)));
    let mut a: f64 = 0.0;
    let mut b: f64 = xyc
        .iter()
        .map(|&(x, y, c)| c * distance(x, y, 0.0, 0.0))
        .fold(0.0, f64::max);
    while b - a > EPS / 2.0 {
        let t = (a + b) / 2.0;
        if check(&xyc, t, k) {
            b = t;
        } else {
            a = t;
        }
    }
    println!("{}", (a + b) / 2.0);
}

fn distance(x0: f64, y0: f64, x1: f64, y1: f64) -> f64 {
    let dx = x1 - x0;
    let dy = y1 - y0;
    (dx * dx + dy * dy).sqrt()
}

fn check(xyc: &[(f64, f64, f64)], t: f64, k: usize) -> bool {
    let n = xyc.len();
    let mut candidates = Vec::with_capacity(n * n);
    for i in 0..n {
        let (xi, yi, ci) = xyc[i];
        let ri = t / ci;
        candidates.push((xi, yi));
        for j in i + 1..n {
            let (xj, yj, cj) = xyc[j];
            let rj = t / cj;
            let d = distance(xi, yi, xj, yj);
            let cos = (d * d + ri * ri - rj * rj) / (2.0 * d * ri);
            if cos * cos <= 1.0 {
                let sin = (1.0 - cos * cos).sqrt();
                let dx = xj - xi;
                let dy = yj - yi;
                let x = xi + ri / d * (cos * dx - sin * dy);
                let y = yi + ri / d * (cos * dy + sin * dx);
                candidates.push((x, y));
                let x = xi + ri / d * (cos * dx + sin * dy);
                let y = yi + ri / d * (cos * dy - sin * dx);
                candidates.push((x, y));
            }
        }
    }
    for (x0, y0) in candidates {
        let mut cnt = 0;
        for &(x, y, c) in xyc {
            if distance(x0, y0, x, y) * c <= t + EPS / 2.0 {
                cnt += 1;
            }
        }
        if cnt >= k {
            return true;
        }
    }
    return false;
}