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)

use std::io::Read;
use std::ops::*;
use std::fmt;

const EPS: f64 = 1e-7;

#[derive(Clone, Copy)]
struct Point(f64, f64);

impl Point {
    fn norm(&self) -> f64 {
        (self.0.powi(2) + self.1.powi(2)).sqrt()
    }
}

impl Add for Point {
    type Output = Point;
    fn add(self, rhs: Self) -> Self::Output {
        Point(self.0 + rhs.0, self.1 + rhs.1)
    }
}

impl Sub for Point {
    type Output = Point;
    fn sub(self, rhs: Self) -> Self::Output {
        Point(self.0 - rhs.0, self.1 - rhs.1)
    }
}

impl fmt::Display for Point {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(f, "({:.3}, {:.3})", self.0, self.1)
    }
}

#[derive(Clone, Copy)]
struct Circle(Point, f64);

impl Circle {
    fn intersect(self, rhs: Self) -> bool {
        let d = (self.0 - rhs.0).norm();
        self.1 + rhs.1 > d + EPS && self.1 + d > rhs.1 + EPS && rhs.1 + d > self.1 + EPS
    }
    fn common(self, rhs: Self) -> (Point, Point) {
        assert!(self.intersect(rhs));
        let p = rhs.0 - self.0;
        let r1 = self.1;
        let r2 = rhs.1;
        let (x, y, z) = (2f64 * p.0, 2f64 * p.1, r1.powi(2) + p.0.powi(2) + p.1.powi(2) - r2.powi(2));
        let (a, b, c) = (x.powi(2) + y.powi(2), -2f64 * x * z, z.powi(2) - (r1 * y).powi(2));
        let det = b * b - 4f64 * a * c;
        let sqdet = det.sqrt();
        let xs = ((-b + sqdet) / (2.0 * a), (-b - sqdet) / (2.0 * a));
        let (a, b, c) = (x.powi(2) + y.powi(2), -2f64 * y * z, z.powi(2) - (r1 * x).powi(2));
        let det = b * b - 4f64 * a * c;
        let sqdet = det.sqrt();
        let mut ys = ((-b - sqdet) / (2.0 * a), (-b + sqdet) / (2.0 * a));
        if (Point(xs.0, ys.0).norm() - r1).abs() > EPS {
            ys = (ys.1, ys.0);
        }
        let a = Point(xs.0, ys.0) + self.0;
        let b = Point(xs.1, ys.1) + self.0;
        (a, b)
    }
    fn contains(&self, p: Point) -> bool {
        (p - self.0).norm() + EPS < self.1
    }
}

impl fmt::Display for Circle {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(f, "({}, {:.3})", self.0, self.1)
    }
}

#[derive(Clone, Copy)]
struct Niku(Point, f64);

impl Niku {
    fn rad(&self, time: f64) -> f64 {
        time / self.1
    }
    fn to_circle(&self, time: f64) -> Circle {
        Circle(self.0, self.rad(time))
    }
}

fn run() {
    let mut s = String::new();
    std::io::stdin().read_to_string(&mut s).unwrap();
    let mut it = s.trim().split_whitespace();
    let n: usize = it.next().unwrap().parse().unwrap();
    let m: usize = it.next().unwrap().parse().unwrap();
    let niku: Vec<Niku> = (0..n).map(|_| {
        let x: f64 = it.next().unwrap().parse().unwrap();
        let y: f64 = it.next().unwrap().parse().unwrap();
        let c: f64 = it.next().unwrap().parse().unwrap();
        Niku(Point(x, y), c)
    }).collect();
    let mut ng = 0f64;
    let mut ok = 2000f64 * 100f64;
    for _ in 0..53 {
        let mid = (ng + ok) * 0.5;
        let p: Vec<_> = niku.iter().map(|a| a.to_circle(mid)).collect();
        let mut valid = false;
        for (i, &x) in p.iter().enumerate() {
            for (j, &y) in p[..i].iter().enumerate() {
                if !x.intersect(y) {
                    continue;
                }
                let (a, b) = x.common(y);
                for &t in [a, b].iter() {
                    let mut cnt = 2;
                    for (k, &z) in p.iter().enumerate() {
                        if k != i && k != j && z.contains(t) {
                            cnt += 1;
                        }
                    }
                    if cnt >= m {
                        valid = true;
                    }
                }
            }
        }
        for (i, &x) in p.iter().enumerate() {
            let mut cnt = 1;
            let x = x.0;
            for (j, &y) in p.iter().enumerate() {
                if j != i && y.contains(x) {
                    cnt += 1;
                }
            }
            if cnt >= m {
                valid = true;
            }
        }
        if valid {
            ok = mid;
        } else {
            ng = mid;
        }
    }
    println!("{:.7}", ok);
}

fn main() {
    run();
}