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)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
use std::io::{Write, BufWriter};
// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

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

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

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

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

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

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

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

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

#[allow(unused)]
macro_rules! debug {
    ($($format:tt)*) => (write!(std::io::stderr(), $($format)*).unwrap());
}
#[allow(unused)]
macro_rules! debugln {
    ($($format:tt)*) => (writeln!(std::io::stderr(), $($format)*).unwrap());
}

const EPS: f64 = 1.0e-9;

// http://is2011.2-d.jp/moin/moin.cgi/amylase/icpc#A.2BUYYwaHb0fdowbk6kcLk-
fn cc_intersect((cx, cy, cr): (f64, f64, f64), (dx, dy, dr) : (f64, f64, f64))
                -> Vec<(f64, f64)> {
    let dist = ((cx - dx) * (cx - dx) + (cy - dy) * (cy - dy)).sqrt();
    if dist > cr + dr + EPS {
        return vec![];
    }
    if dist < (cr - dr).abs() - EPS {
        return vec![];
    }
    let s = (cr + dr + dist) / 2.0;
    let area = (s * (s - cr) * (s - dr) * (s - dist)).sqrt();
    let h = 2.0 * area / dist;
    let (vx, vy) = (dx - cx, dy - cy);
    let vabs = (vx * vx + vy * vy).sqrt();
    let (vx, vy) = (vx / vabs, vy / vabs);
    let ss = (cr * cr - h * h).sqrt();
    let (mx, my) = (cx + ss * vx, cy + ss * vy);
    let (nx, ny) = (-vy, vx);
    vec![(mx + nx * h, my + ny * h), (mx - nx * h, my - ny * h)]
}

fn solve() {
    let out = std::io::stdout();
    let mut out = BufWriter::new(out.lock());
    macro_rules! puts {
        ($($format:tt)*) => (let _ = write!(out,$($format)*););
    }
    input! {
        n: usize, k: usize,
        xyc: [(f64, f64, f64); n],
    }
    if n <= 1 {
        puts!("0.0\n");
        return;
    }
    let mut pass = 1000000.0;
    let mut fail = 0.0;
    for _ in 0..100 {
        let mid = (pass + fail) / 2.0;
        let mut pts = vec![];
        let mut xyr = xyc.clone();
        for i in 0..n {
            xyr[i].2 = mid / xyc[i].2;
        }
        for i in 0..n {
            pts.push((xyr[i].0, xyr[i].1));
        }
        for i in 0..n {
            for j in 0..i {
                pts.append(&mut cc_intersect(xyr[i], xyr[j]));
            }
        }
        let mut ok = false;
        for &(x, y) in &pts {
            let mut count = 0;
            for i in 0..n {
                let (cx, cy, cr) = xyr[i];
                let dist = (x - cx) * (x - cx) + (y - cy) * (y - cy);
                if dist < cr * cr + EPS {
                    count += 1;
                }
            }
            if count >= k {
                ok = true;
                break;
            }
        }
        if ok {
            pass = mid;
        } else {
            fail = mid;
        }
    }
    puts!("{}\n", pass);
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}