Holes

AtCoder

IDagc021_b

Time2000ms

Memory256MB

Difficulty—

There are $N$ holes in a two-dimensional plane. The coordinates of the $i$\-th hole are $(x_i,y_i)$. Let $R=10^{10^{10^{10}}}$. Ringo performs the following operation: * Randomly choose a point from the interior of a circle of radius $R$ centered at the origin, and put Snuke there. Snuke will move to the hole with the smallest Euclidean distance from the point, and fall into that hole. If there are multiple such holes, the hole with the smallest index will be chosen. For every $i$ $(1 \leq i \leq N)$, find the probability that Snuke falls into the $i$\-th hole. Here, the operation of randomly choosing a point from the interior of a circle of radius $R$ is defined as follows: * Pick two real numbers $x$ and $y$ independently according to uniform distribution on $[-R,R]$. * If $x^2+y^2\leq R^2$, the point $(x,y)$ is chosen. Otherwise, repeat picking the real numbers $x,y$ until the condition is met. ## Constraints * $2 \leq N \leq 100$ * $|x_i|,|y_i| \leq 10^6(1\leq i\leq N)$ * All given points are pairwise distinct. * All input values are integers. ## Input Input is given from Standard Input in the following format: $N$ $x_1$ $y_1$ $:$ $x_N$ $y_N$ [samples]

Samples

Input #1

2
0 0
1 1

Output #1

0.5
0.5

If Ringo put Snuke in the region $x+y\leq 1$, Snuke will fall into the first hole. The probability of this happening is very close to $0.5$. Otherwise, Snuke will fall into the second hole, the probability of which happening is also very close to $0.5$.

Input #2

Output #2

0.43160120892732328768
0.03480224363653196956
0.13880483535586193855
0.00000000000000000000
0.39479171208028279727

API Response (JSON)

{
  "problem": {
    "name": "Holes",
    "description": {
      "content": "There are $N$ holes in a two-dimensional plane. The coordinates of the $i$\\-th hole are $(x_i,y_i)$. Let $R=10^{10^{10^{10}}}$. Ringo performs the following operation: *   Randomly choose a point fro",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc021_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ holes in a two-dimensional plane. The coordinates of the $i$\\-th hole are $(x_i,y_i)$.\nLet $R=10^{10^{10^{10}}}$. Ringo performs the following operation:\n\n*   Randomly choose a point fro...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments