Question 57
Let $n$ be a positive integer. Let $S$ be the set of points in the coordinate plane whose $x$- and $y$-coordinates are both non-negative integers less than $2n$ (so $S$ contains $4n^2$ points). Suppose $\mathcal{F}$ is a set of $n^2$ quadrilaterals such that all vertices of every quadrilateral lie in $S$, and every point of $S$ is a vertex of one of the quadrilaterals in $\mathcal{F}$. Determine the largest possible value of the sum of the areas of all $n^2$ quadrilaterals in $\mathcal{F}$.