Yakiniku Optimization Problem

AtCoder

IDabc157_f

Time2000ms

Memory256MB

Difficulty—

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: $N$ $K$ $x_1$ $y_1$ $c_1$ $\vdots$ $x_N$ $y_N$ $c_N$ [samples]

Samples

Input #1

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.

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

Output #2

7411.2252

API Response (JSON)

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      "content": "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",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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_translate": false,
      "language": "English"
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}

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