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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

#!/usr/bin/env python3

import sys
import numpy as np
import math
from collections import namedtuple
from itertools import combinations

DEBUG = False

def read(t):
    return t(sys.stdin.readline().rstrip())


def read_list(t, sep = " "):
    return [t(s) for s in sys.stdin.readline().rstrip().split(sep)]


def dprint(*args, **kwargs):
    if DEBUG:
        print(*args, **kwargs)
    return

Eps = 1e-9
Pos = namedtuple("Pos", ("y", "x"))
Meat = namedtuple("Meat", ("c", "pos"))
Circle = namedtuple("Circle", ("r", "center"))

def feq(f1, f2):
    return abs(f1 - f2) < Eps

def intersection_points(circle1, circle2):
    v = circle2.center - circle1.center
    v_l = np.linalg.norm(v, ord=2)
    if feq(v_l, 0):
        return []
    if feq(v_l + circle1.r + circle2.r, max(v_l, circle1.r, circle2.r) * 2):
        return [circle1.center + v * (circle1.r / v_l)]
    if v_l + circle1.r + circle2.r < max(v_l, circle1.r, circle2.r) * 2:
        return []
    x = -(circle2.r**2 - v_l**2 - circle1.r**2) / (2 * v_l)
    y = math.sqrt(circle1.r**2 - x**2)
    mid = circle1.center + v * (x / v_l)
    v = np.array([-v[1], v[0]])
    return [mid + v * (y / v_l), mid - v * (y / v_l)]

def can_cook_meats(time, meats, nr_to_cook):
    cookable_areas = [Circle(time / meat.c, np.array([meat.pos.y, meat.pos.x])) for meat in meats]

    heat_candidates = []
    for area in cookable_areas:
        heat_candidates.append(area.center)
    for a1, a2 in combinations(cookable_areas, 2):
        heat_candidates += intersection_points(a1, a2)
    
    for heat_candidate in heat_candidates:
        cookable_cnt = 0
        for area in cookable_areas:
            if np.linalg.norm(heat_candidate - area.center, ord=2) < area.r + Eps:
                cookable_cnt += 1
        if cookable_cnt >= nr_to_cook:
            return True
    return False


def main():
    n, k = read_list(int)
    meats = []
    for _ in range(n):
        x, y, c = read_list(int)
        meats.append(Meat(c, Pos(y, x)))
    
    begin = 0
    end = 15e4  # sqrt(2) * 1e3 * 1e2
    t = begin
    # Binary search on real number a constant times
    for _ in range(50): 
        if can_cook_meats(t, meats, k):
            end = t
            t = (begin + t) / 2
        else:
            begin = t
            t = (t + end) / 2
    print(t)


if __name__ == "__main__":
    main()