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: Wrong Answer)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use competitive::geo::*;
use competitive::prelude::*;

const EPS: f64 = 1e-9;

fn main() {
    input! {
        n: usize,
        k: usize,
        qs: [(f64, f64, f64); n],
    }

    let ans = binary_search(0.0, 100.0 * 2000.0, |t| {
        let mut cross_points = vec![];
        for i in 0..n {
            let p = Complex::new(qs[i].0, qs[i].1);
            let pr = t / qs[i].2;
            for j in i + 1..n {
                let q = Complex::new(qs[j].0, qs[j].1);
                let qr = t / qs[j].2;

                if let Some((c1, c2)) = cross_point(p, pr, q, qr) {
                    cross_points.push(c1);
                    cross_points.push(c2);
                }
            }
        }

        let mut best = 1;

        for c in cross_points {
            let mut cur = 0;
            for i in 0..n {
                let p = Complex::new(qs[i].0, qs[i].1);
                let r = t / qs[i].2;

                if (p - c).norm() <= r + EPS {
                    cur += 1;
                }
            }
            best = max(best, cur);
        }

        best < k
    });
    echo!(ans);
}

fn cross_point(a: Pt, ar: f64, b: Pt, br: f64) -> Option<(Pt, Pt)> {
    let c = (a - b).norm();

    if c >= ar + br || c + br <= ar || c + ar <= br {
        return None;
    }

    let x = (ar.powi(2) - br.powi(2) + c.powi(2)) / (2.0 * c);
    let h = (ar.powi(2) - x.powi(2)).sqrt();

    let v = (b - a) / c;
    let c1 = a + x * v + h * v * Pt::from_polar(&1.0, &(std::f64::consts::PI / 2.0));
    let c2 = a + x * v - h * v * Pt::from_polar(&1.0, &(std::f64::consts::PI / 2.0));

    let y = c - x;

    // dbg!(ar, br, c, x, h, ar);
    // dbg!(y.powi(2) + h.powi(2) - br.powi(2));
    // dbg!((c1 - a).norm() - ar);
    // dbg!((c2 - b).norm() - br);

    assert!(((x.powi(2) + h.powi(2)).sqrt() - ar).abs() < EPS);
    assert!(((y.powi(2) + h.powi(2)).sqrt() - br).abs() < EPS);
    assert!(((c1 - a).norm() - ar).abs() < EPS);
    assert!(((c1 - b).norm() - br).abs() < EPS);
    assert!(((c2 - a).norm() - ar).abs() < EPS);
    assert!(((c2 - b).norm() - br).abs() < EPS);

    Some((c1, c2))
}
*/

fn main() {
    let exe = "/tmp/bin2EDF2A12";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";