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, unused_macros, dead_code)]

macro_rules! min {
    (.. $x:expr) => {{
        let mut it = $x.iter();
        it.next().map(|z| it.fold(z, |x, y| min!(x, y)))
    }};
    ($x:expr) => ($x);
    ($x:expr, $($ys:expr),*) => {{
        let t = min!($($ys),*);
        if $x < t { $x } else { t }
    }}
}
macro_rules! max {
    (.. $x:expr) => {{
        let mut it = $x.iter();
        it.next().map(|z| it.fold(z, |x, y| max!(x, y)))
    }};
    ($x:expr) => ($x);
    ($x:expr, $($ys:expr),*) => {{
        let t = max!($($ys),*);
        if $x > t { $x } else { t }
    }}
}
macro_rules! ewriteln {
    ($($args:expr),*) => { let _ = writeln!(&mut std::io::stderr(), $($args),*); };
}
macro_rules! trace {
    ($x:expr) => { ewriteln!(">>> {} = {:?}", stringify!($x), $x) };
    ($($xs:expr),*) => { trace!(($($xs),*)) }
}
macro_rules! put {
    (.. $x:expr) => {{
        let mut it = $x.iter();
        if let Some(x) = it.next() { print!("{}", x); }
        for x in it { print!(" {}", x); }
        println!("");
    }};
    ($x:expr) => { println!("{}", $x) };
    ($x:expr, $($xs:expr),*) => { print!("{} ", $x); put!($($xs),*) }
}

const M: i64 = 1_000_000_007;

trait Group: std::ops::Add<Output=Self> + std::ops::Sub<Output=Self> + std::ops::Neg<Output=Self>
    + std::iter::Sum + Clone + Copy {
    fn zero() -> Self;
}
impl Group for i64 { fn zero() -> Self { 0 }}
impl Group for f64 { fn zero() -> Self { 0.0 }}

trait Ring: Group + std::ops::Mul<Output=Self> + std::ops::Div<Output=Self> {
    fn one() -> Self;
}
impl Ring for i64 { fn one() -> Self { 1 }}
impl Ring for f64 { fn one() -> Self { 1.0 }}

#[derive(PartialEq, Eq, PartialOrd, Ord, Clone, Copy, Debug)]
enum Hyper<X> { NegInf, Real(X), Inf }
use Hyper::{Real, NegInf, Inf};
impl<X: Ring> std::ops::Add for Hyper<X> {
    type Output = Self;
    fn add(self, rhs: Hyper<X>) -> Hyper<X> {
        match (self, rhs) {
            (Real(x), Real(y)) => Real(x + y),
            (Inf, _) => Inf,
            (_, Inf) => Inf,
            _ => NegInf,
        }
    }
}
impl<X: Ring> std::ops::Sub for Hyper<X> {
    type Output = Self;
    fn sub(self, rhs: Hyper<X>) -> Hyper<X> {
        self + (-rhs)
    }
}
impl<X: Ring> std::ops::Neg for Hyper<X> {
    type Output = Self;
    fn neg(self) -> Hyper<X> {
        match self {
            Inf => NegInf,
            NegInf => Inf,
            Real(x) => Real(-x),
        }
    }
}

/// Thanks to: https://qiita.com/hatoo@github/items/fa14ad36a1b568d14f3e
#[derive(Copy, Clone, Debug, PartialEq, PartialOrd)]
struct Total<T>(T);
impl<T: PartialEq> Eq for Total<T> {}
impl<T: PartialOrd> Ord for Total<T> {
    fn cmp(&self, rhs: &Total<T>) -> std::cmp::Ordering {
        self.0.partial_cmp(&rhs.0).unwrap()
    }
}

fn main() {
    let mut sc = Scanner::new();
    let n: usize = sc.cin();
    let k: usize = sc.cin();
    let nikus: Vec<_> = (0..n).map(|_| {
        let x: f64 = sc.cin();
        let y: f64 = sc.cin();
        let c: f64 = sc.cin();
        (x, y, c)
    }).collect();

    let mut ans: Hyper<f64> = Inf;

    let mut cands = vec![];
    for &n in nikus.iter() {
        cands.push((n.0, n.1));
    }
    for i in 0..9 {
        for j in 0..9 {
            cands.push((-1000.0 + (i as f64) * 2000.0 / 8.0,
                        -1000.0 + (j as f64) * 2000.0 / 8.0));
        }
    }

    for &p in cands.iter() {
        let mut x: f64 = p.0;
        let mut y: f64 = p.1;
        let mut gamma = 0.1;
        for _ in 0..4000 {
            gamma *= 0.99;
            let mut scores: Vec<_> = vec![];
            for i in 0..n {
                let niku = nikus[i];
                let s = ((niku.0 - x).powi(2) + (niku.1 - y).powi(2)).sqrt() * niku.2;
                scores.push((Total(s), i));
            }
            scores.sort();
            ans = min!(ans, Real((scores[k-1].0).0));
            let idx = scores[k-1].1;
            x += gamma * (nikus[idx].0 - x);
            y += gamma * (nikus[idx].1 - y);
        }
    }

    if let Real(ans) = ans {
        println!("{:.32}", ans);
    }

}

use std::io::{self, Write};
use std::str::FromStr;
use std::collections::VecDeque;

struct Scanner { stdin: io::Stdin, buffer: VecDeque<String>, }
impl Scanner {
    fn new() -> Self { Scanner { stdin: io::stdin(), buffer: VecDeque::new() } }
    fn cin<T: FromStr>(&mut self) -> T {
        while self.buffer.len() == 0 {
            let mut line = String::new();
            let _ = self.stdin.read_line(&mut line);
            for w in line.split_whitespace() {
                self.buffer.push_back(String::from(w));
            }
        }
        self.buffer.pop_front().unwrap().parse::<T>().ok().unwrap()
    }
    fn chars(&mut self) -> Vec<char> { self.cin::<String>().chars().collect() }
    fn vec<T: FromStr>(&mut self, n: usize) -> Vec<T> { (0..n).map(|_| self.cin()).collect() }
}