Problem: A plane has a special point $O$ called the origin.
Let $P$ be a set of $2021$ points in the plane such that
\begin{itemize}
\ii no three points in $P$ lie on a line and
\ii no two points in $P$ lie on a line through the origin.
\end{itemize}
A triangle with vertices in $P$ is \emph{fat}
if $O$ is strictly inside the triangle.
Find the maximum number of fat triangles.

Solution: For every pair of points $P$ and $Q$,
draw a directed arrow $P \to Q$ if $\angle POQ$ is labeled clockwise.
This gives a tournament on $2021$ vertices,
and a triangle is fat if its three edges form a directed cycle.

Consequently, by
\href{https://aops.com/community/c6h130647p740398}{\textbf{Canada 2006/4}},
there are at most $\frac16 n(n+1)(2n+1)$
fat triangles, where $n = 1010$.
Equality occurs if the set of points are the vertices of a regular
$2021$-gon containing $O$.