EGMO 2021 Problem 5

A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that (i) no three points in $P$ lie on a line and (ii) no two points in $P$ lie on a line through the origin.

A triangle with vertices in $P$ is fat, if $O$ is strictly inside the triangle. Find the maximum number of fat triangles. (Austria, Veronika Schreitter)