Integer Convex Hull

AtCoder

IDabc202_f

Time2000ms

Memory256MB

Difficulty—

We have $N$ points $P_1, P_2, \dots, P_N$ on a plane. The coordinates of $P_i$ is $(X_i, Y_i)$. We know that no three points lie on the same line. For a subset $S$ of ${ P_1, P_2, \dots, P_N }$ with at least three elements, let us define the **convex hull** of $S$ as follows: * The convex hull is the convex polygon with the minimum area such that every point of $S$ is within or on the circumference of that polygon. Find the number, modulo $(10^9 + 7)$, of subsets $S$ such that the area of the convex hull is an integer. ## Constraints * $3 \leq N \leq 80$ * $0 \leq |X_i|, |Y_i| \leq 10^4$ * No three points lie on the same line. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $X_1$ $Y_1$ $X_2$ $Y_2$ $\vdots$ $X_N$ $Y_N$ [samples]

Samples

Input #1

Output #1

2

${ P_1, P_2, P_4 }$ and ${ P_2, P_3, P_4 }$ satisfy the condition.

Input #2

23
-5255 7890
5823 7526
5485 -113
7302 5708
9149 2722
4904 -3918
8566 -3267
-3759 2474
-7286 -1043
-1230 1780
3377 -7044
-2596 -6003
5813 -9452
-9889 -7423
2377 1811
5351 4551
-1354 -9611
4244 1958
8864 -9889
507 -8923
6948 -5016
-6139 2769
4103 9241

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Integer Convex Hull",
    "description": {
      "content": "We have $N$ points $P_1, P_2, \\dots, P_N$ on a plane. The coordinates of $P_i$ is $(X_i, Y_i)$. We know that no three points lie on the same line. For a subset $S$ of ${ P_1, P_2, \\dots, P_N }$ with a",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
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    "is_sync": true,
    "sync_url": null,
    "sign": "abc202_f"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have $N$ points $P_1, P_2, \\dots, P_N$ on a plane. The coordinates of $P_i$ is $(X_i, Y_i)$. We know that no three points lie on the same line.\nFor a subset $S$ of ${ P_1, P_2, \\dots, P_N }$ with a...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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