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)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

#[derive(PartialEq, PartialOrd, Clone, Copy)]
pub struct Total<T>(pub T);

impl<T: PartialEq> Eq for Total<T> {}

impl<T: PartialOrd> Ord for Total<T> {
    fn cmp(&self, other: &Total<T>) -> std::cmp::Ordering {
        self.0.partial_cmp(&other.0).unwrap()
    }
}

fn grid_search(
    center_x: f64,
    center_y: f64,
    step: f64,
    k: usize,
    niku: &Vec<(f64, f64, f64)>,
) -> ((f64, f64), f64) {
    let mut ans = (0.0, 0.0);
    let mut cur_best = std::f64::MAX;
    let mut time = vec![Total(0.0); niku.len()];
    for xi in -110..111 {
        for yi in -110..111 {
            let xt = center_x + xi as f64 * step;
            let yt = center_y + yi as f64 * step;
        }
    }
    (ans, cur_best)
}

fn main() {
    let v = read_vec::<usize>();
    let (n, k) = (v[0], v[1]);
    let mut niku = vec![];
    for i in 0..n {
        let v = read_vec::<i64>();
        niku.push((v[0] as f64, v[1] as f64, v[2] as f64));
    }
    let mut ans = std::f64::MAX;

    for i in 0..n {
        for j in i + 1..n {
            let (x1, y1, c1) = niku[i];
            let (x2, y2, c2) = niku[j];
            let xc = (x1 * c2 + x2 * c1) / (c1 + c2);
            let yc = (y1 * c2 + y2 * c1) / (c1 + c2);
            let vv = (
                y1 - y2 / (x1 - x2).hypot(y1 - y2),
                x1 - x2 / (x1 - x2).hypot(y1 - y2),
            );
            let lb = ternary_search(-2000.0, 2000.0, |d| {
                let mut time = vec![Total(0.0); n];
                let xt = xc + d * vv.0;
                let yt = yc + d * vv.1;

                for (i, &(x, y, c)) in niku.iter().enumerate() {
                    time[i] = Total(c * (xt - x).hypot(yt - y));
                }
                time[k - 1].0
            });
            if ans > lb {
                ans = lb;
            }
        }
    }
    println!("{}", ans);
}
fn ternary_search<F>(lb: f64, ub: f64, criterion: F) -> f64
where
    F: Fn(f64) -> f64,
{
    let mut lb = lb;
    let mut ub = ub;
    let mid = (lb + ub) / 2.;
    let mut convex_upward = false;
    if criterion(mid) > (criterion(lb) + criterion(ub)) / 2. {
        convex_upward = true;
    }
    while ub - lb > 0.001 {
        let mid1 = (ub + lb * 2.) / 3.;
        let mid2 = (ub * 2. + lb) / 3.;
        if convex_upward {
            if criterion(mid1) < criterion(mid2) {
                lb = mid1;
            } else {
                ub = mid2;
            }
        } else {
            if criterion(mid1) > criterion(mid2) {
                lb = mid1;
            } else {
                ub = mid2;
            }
        }
    }
    (lb + ub) / 2.
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}