Score : $500$ points

问题描述

ABC 250 对于目标举办 ABC 1000 的高桥来说是一个值得纪念的四分之一里程碑，因此他打算通过尽可能接近吃掉所买披萨的 $1/4$ 来庆祝这次比赛。

高桥购买的披萨呈平面凸 $N$ 边形形状。当披萨放在 $xy$ 平面上时，第 $i$ 个顶点的坐标为 $(X_i, Y_i)$。

高桥决定按照以下方式切割并食用披萨：

• 首先，高桥从披萨的顶点中选择两个不相邻的顶点，并沿过这两点的直线用刀进行切割，将披萨分割成两块。
• 然后，他任意选择其中一块食用。

令 $a$ 为高桥所购披萨面积的四分之一（$=1/4$），$b$ 为高桥所吃披萨的那一块面积。求 $8 \times |a-b|$ 可能达到的最小值。可以证明这个值始终为整数。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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:

$N$

$X_1$ $Y_1$

$X_2$ $Y_2$

$\dots$

$X_N$ $Y_N$

Output

Print the answer as an integer.

Sample Input 1

5
3 0
2 3
-1 3
-3 1
-1 -1

Sample 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.

Sample Input 2

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

Sample Output 2

1280000000000000000

Sample Input 3

6
-816 222
-801 -757
-165 -411
733 131
835 711
-374 979

Sample Output 3

157889

update @ 2024/3/10 10:41:39