19419번 - Split Game 다국어

Consider the following game about splitting a simple polygon with $N$ vertices on a plane. The purpose of this game is using a straight line which passes through the origin to split the given simple polygon into as many non-zero area regions as possible. Please finish the game with the best result possible.

The input consists of $N+1$ lines. The first line contains an integer $N$. The $i$-th of the following $N$ lines consists of two integers $x_i$ and $y_i$ indicating the vertices of the given polygon in counter-clockwise order.

Output one integer: the maximum number of non-zero area regions into which the given polygon can be split by a single line passing through the origin.

• $1 \le N \le 10^5$
• $1 \le x_i, y_i \le 10^9$
• if $i \ne j$, then $(x_i, y_i) \ne (x_j, y_j)$
• the vertices are given in counter-clockwise order