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 PyPy3 (2.4.0) (judge verdict: Wrong Answer)

import math
import random
import time
stime = time.time()
N, K = map(int, input().split())
X = []
for _ in range(N):
    x, y, c = map(int, input().split())
    X.append((x, y, c))

def f(x0, y0):
    L = []
    for x, y, c in X:
        L.append(math.sqrt((x-x0)**2+(y-y0)**2) * c)
    return sorted(L)[K-1]


mi = 1 << 100
for i, (x1, y1, _) in enumerate(X):
    for x2, y2, __ in X[:i]:
        x0, y0 = (x1+x2) / 2, (y1+y2) / 2
        a = f(x0, y0)
        mi = min(mi, a)
        r = 100
        k = 0.7
        for i in range(300):
            dx, dy = random.randrange(-10, 11) * r, random.randrange(-10, 11) * r
            nx0, ny0 = x0 + dx, y0 + dy
            b = f(nx0, ny0)
            if a > b:
                mi = min(mi, b)
                x0, y0 = nx0, ny0
                a = b
                k = 0.95
            r *= k
    if time.time() - stime > 1.75:
        break

for x1, y1, _ in X:
    if time.time() - stime > 1.75:
        break
    for x2, y2, __ in X:
        x0, y0 = (x1+x2*2) / 3, (y1+y2*2) / 3
        a = f(x0, y0)
        mi = min(mi, a)
        r = 100
        k = 0.7
        for i in range(300):
            dx, dy = random.randrange(-10, 11) * r, random.randrange(-10, 11) * r
            nx0, ny0 = x0 + dx, y0 + dy
            b = f(nx0, ny0)
            if a > b:
                mi = min(mi, b)
                x0, y0 = nx0, ny0
                a = b
                k = 0.95
            r *= k
print(mi)