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: Accepted)

import math

n, k = map(int, input().split())
x, y, c = zip(*[map(int, input().split()) for _ in range(n)])
que = [(0, 0, 1000)]
res = 1e9
i = 0
while i < len(que):
    px, py, r = que[i]
    i += 1
    lb = [c[j] * math.hypot(max(abs(px - x[j]) - r, 0), max(abs(py - y[j]) - r, 0)) for j in range(n)]
    if res * (1 - 5e-7) > sorted(lb)[k - 1]:
        dis = [c[j] * math.hypot(px - x[j], py - y[j]) for j in range(n)]
        res = min(res, sorted(dis)[k - 1])
        for dx in (-0.5, 0.5):
            for dy in (-0.5, 0.5):
                que.append((px + dx * r, py + dy * r, r / 2))
print(res)