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::{Ordering};

#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};

#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};

#[allow(unused_imports)]
use std::iter::FromIterator;

#[allow(unused_macros)]
macro_rules! min {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::min($x, min!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! max {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::max($x, max!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! get { 
    ([$t: ty]) => { 
        { 
            let mut line = String::new(); 
            stdin().read_line(&mut line).unwrap(); 
            line.split_whitespace().map(|t|t.parse::<$t>().unwrap()).collect::<Vec<_>>()
        }
    };
    ([$t: ty];$n: expr) => {
        (0..$n).map(|_|get!([$t])).collect::<Vec<_>>()
    };
    ($t: ty) => {
        {
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            line.trim().parse::<$t>().unwrap()
        }
    };
    ($($t: ty),*) => {
        { 
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            let mut iter = line.split_whitespace();
            ($(iter.next().unwrap().parse::<$t>().unwrap(),)*)
        }
    };
    ($t: ty; $n: expr) => {
        (0..$n).map(|_|get!($t)).collect::<Vec<_>>()
    };
    ($($t: ty),*; $n: expr) => {
        (0..$n).map(|_|get!($($t),*)).collect::<Vec<_>>()
    };
}

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

const BIG_STACK_SIZE: bool = true;

#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}

fn ternary_search<T, F>(l: f64, r: f64, crit: &F) -> f64
where
    T: std::cmp::PartialOrd,
    F: Fn(f64) -> T,
{
    let mut l = l;
    let mut r = r;
    for _ in 0..100 {
        let nl = l + (r - l) / 3.0;
        let nr = l + 2.0 * (r - l) / 3.0;
        if crit(nl) > crit(nr) {
            l = nl;
        } else {
            r = nr;
        }
    }
    l
}

#[derive(PartialEq, PartialOrd, Copy, Clone)]
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()
    }
}
impl<T: std::str::FromStr> std::str::FromStr for Total<T> {
    type Err = T::Err;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let res = T::from_str(s)?;
        Ok(Total(res))
    }
}
impl<T: std::fmt::Display> std::fmt::Debug for Total<T> {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}
impl<T: std::fmt::Display> std::fmt::Display for Total<T> {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}
impl<T: std::ops::Add<Output = T>> std::ops::Add for Total<T> {
    type Output = Self;
    fn add(self, other: Self) -> Self {
        Total(self.0 + other.0)
    }
}
impl<T: std::ops::Sub<Output = T>> std::ops::Sub for Total<T> {
    type Output = Self;
    fn sub(self, other: Self) -> Self {
        Total(self.0 - other.0)
    }
}
impl<T: std::ops::Mul<Output = T>> std::ops::Mul for Total<T> {
    type Output = Self;
    fn mul(self, other: Self) -> Self {
        Total(self.0 * other.0)
    }
}
impl<T: std::ops::Div<Output = T>> std::ops::Div for Total<T> {
    type Output = Self;
    fn div(self, other: Self) -> Self {
        Total(self.0 / other.0)
    }
}
impl<T: std::ops::Neg<Output = T>> std::ops::Neg for Total<T> {
    type Output = Self;
    fn neg(self) -> Self::Output {
        Total(-self.0)
    }
}

fn solve() {
    let (n,k) = get!(usize, usize);
    let xyc  = get!(f64,f64,f64; n);

    let time = |xl: f64, yl:f64| -> f64 {
        let mut vec =  Vec::new();
        for &(x,y,c) in &xyc {
            vec.push(c*((x-xl).powf(2.0)+(y-yl).powf(2.0)).sqrt());
        }
        vec.sort_by_key(|&e| Total(e));
        vec[k-1]
    };

    let crit = |x: f64| -> f64 {
        let crit =  |y: f64| -> f64 {
            time(x,y)
        };
        let y = ternary_search(-2000.0,2000.0,&crit);
        time(x,y)
    };
    let x = ternary_search(-2000.0,2000.0,&crit);
    let crit =  |y: f64| -> f64 {
        time(x,y)
    };
    let y = ternary_search(-2000.0,2000.0,&crit);
    println!("{}", time(x,y));
}