One Fourth

AtCoder

IDabc250_f

Time2000ms

Memory256MB

Difficulty—

ABC 250 is a commemorable quarter milestone for Takahashi, who aims to hold ABC 1000, so he is going to celebrate this contest by eating as close to $1/4$ of a pizza he bought as possible. The pizza that Takahashi bought has a planar shape of convex $N$\-gon. When the pizza is placed on an $xy$\-plane, the $i$\-th vertex has coordinates $(X_i, Y_i)$. Takahashi has decided to cut and eat the pizza as follows. * First, Takahashi chooses two non-adjacent vertices from the vertices of the pizza and makes a cut with a knife along the line passing through those two points, dividing the pizza into two pieces. * Then, he chooses one of the pieces at his choice and eats it. Let $a$ be the quarter ($=1/4$) of the area of the pizza that Takahashi bought, and $b$ be the area of the piece of pizza that Takahashi eats. Find the minimum possible value of $8 \times |a-b|$. We can prove that this value is always an integer. ## Constraints * All values in input are integers. * $4 \le N \le 10^5$ * $|X_i|, |Y_i| \le 4 \times 10^8$ * The given points are the vertices of a convex $N$\-gon in the counterclockwise order. ## Input Input is given from Standard Input in the following format: $N$ $X_1$ $Y_1$ $X_2$ $Y_2$ $\dots$ $X_N$ $Y_N$ [samples]

Samples

Input #1

Output #1

1

Suppose that he makes a cut along the line passing through the $3$\-rd and the $5$\-th vertex and eats the piece containing the $4$\-th vertex.  
Then, $a=\frac{33}{2} \times \frac{1}{4} = \frac{33}{8}$, $b=4$, and $8 \times |a-b|=1$, which is minimum possible.

Input #2

4
400000000 400000000
-400000000 400000000
-400000000 -400000000
400000000 -400000000

Output #2

1280000000000000000

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "One Fourth",
    "description": {
      "content": "ABC 250 is a commemorable quarter milestone for Takahashi, who aims to hold ABC 1000, so he is going to celebrate this contest by eating as close to $1/4$ of a pizza he bought as possible. The pizza t",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc250_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "ABC 250 is a commemorable quarter milestone for Takahashi, who aims to hold ABC 1000, so he is going to celebrate this contest by eating as close to $1/4$ of a pizza he bought as possible.\nThe pizza t...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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