Two Convex Sets

AtCoder

IDttpc2024_1_h

Time4000ms

Memory256MB

Difficulty—

A set of points $U$ in the $xy$\-plane is called **good** if no point in $U$ lies in the **interior** of the convex hull of $U$. Note that the empty set is considered good. You are given $N$ distinct points $v_1, v_2, \dots, v_N$ in the $xy$\-plane. The coordinates of the point $v_i$ are $(x_i, y_i)$. No three distinct points are collinear. Count the number of subsets $S$ of $V = \lbrace v_1, v_2, \dots, v_N \rbrace$ such that both $S$ and $V \setminus S$ are good sets. ## Constraints * All input values are integers. * $1 \le N \le 40$ * $|x_i|,|y_i|\le 10^6$ * $(x_i,y_i)\neq (x_j,y_j)\ (i\neq j)$ * No three distinct points are collinear. ## Input The input is given in the following format: $N$ $x_1$ $y_1$ $x_2$ $y_2$ $\vdots$ $x_N$ $y_N$ [samples]

Samples

Input #1

Output #1

14

Except for $\emptyset$ and $V$, all other sets $S$ satisfy the condition.

Input #2

Output #2

Input #3

Output #3

API Response (JSON)

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    "name": "Two Convex Sets",
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      "content": "A set of points $U$ in the $xy$\\-plane is called **good** if no point in $U$ lies in the **interior** of the convex hull of $U$. Note that the empty set is considered good. You are given $N$ distinct ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
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      "memory_limit": 262144
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    "difficulty": "None",
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    "sign": "ttpc2024_1_h"
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      "statement_type": "Markdown",
      "content": "A set of points $U$ in the $xy$\\-plane is called **good** if no point in $U$ lies in the **interior** of the convex hull of $U$. Note that the empty set is considered good.\nYou are given $N$ distinct ...",
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