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)

N,K = map(int,input().split())
XYC = [tuple(map(int,input().split())) for i in range(N)]

from math import hypot,acos,atan2,cos,sin
def circles_cross_points(x1,y1,r1,x2,y2,r2):
    d = hypot(x1-x2, y1-y2)
    if r1 + r2 < d or abs(r1 - r2) >= d: return []
    x = (r1**2 + d**2 - r2**2) / (2*r1*d)
    if not -1 <= x <= 1: return []
    a = acos(x)
    t = atan2(y2-y1, x2-x1)
    return [
        (x1+cos(t+a)*r1, y1+sin(t+a)*r1),
        (x1+cos(t-a)*r1, y1+sin(t-a)*r1)
    ]

eps = 10e-10
def is_ok(t):
    cands = []
    for x,y,_ in XYC:
        cands.append((x,y))
    for i,(x1,y1,c1) in enumerate(XYC[:-1]):
        for x2,y2,c2 in XYC[i+1:]:
            ps = circles_cross_points(x1,y1,t/c1,x2,y2,t/c2)
            cands += ps
    for cx,cy in cands:
        k = 0
        for x,y,c in XYC:
            d = hypot(cx-x,cy-y)
            if c*d <= t + eps:
                k += 1
                if k >= K:
                    return True
    return False

ok = 10**10
ng = 0
for _ in range(100):
    m = (ok+ng)/2
    if is_ok(m):
        ok = m
    else:
        ng = m
print(ok)