Score : $600$ points

问题描述

Takahashi 想在一个可以视为二维平面的烧烤网上烤制 $N$ 块肉。第 $i$ 块肉的坐标为 $\left(x_i, y_i\right)$，其硬度为 $c_i$。

Takahashi 可以使用一个热源来烧烤肉块。若他将热源置于坐标为 $\left(X, Y\right)$ 的位置，其中 $X$ 和 $Y$ 是实数，则第 $i$ 块肉将在 $c_i \times \sqrt{\left(X - x_i\right)^2 + \left(Y-y_i\right)^2}$ 秒后烤熟。

Takahashi 想要吃到 $K$ 块或更多的熟肉。请计算如果他将热源放置在能够使所需时间最短的位置上，需要多少时间才能烤熟至少 $K$ 块肉。

以上为通义千问 qwen-max 翻译，仅供参考。

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

update @ 2024/3/10 17:15:13