ISL 2021 G3

Version 1. Let $n$ be a fixed positive integer, and let S be the set of points $(x, y)$ on the Cartesian plane such that both coordinates $x$ and $y$ are nonnegative integers smaller than $2 n$ (thus $|S|=4 n^{2}$ ). Assume that $\mathcal{F}$ is a set consisting of $n^{2}$ quadrilaterals such that all their vertices lie in $S$, and each point in $S$ is a vertex of exactly one of the quadrilaterals in $\mathcal{F}$. Determine the largest possible sum of areas of all $n^{2}$ quadrilaterals in $\mathcal{F}$. Version 2. Let $n$ be a fixed positive integer, and let S be the set of points $(x, y)$ on the Cartesian plane such that both coordinates $x$ and $y$ are nonnegative integers smaller than $2 n$ (thus $|\mathrm{S}|=4 n^{2}$ ). Assume that $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in S , and each point in S is a vertex of exactly one of the polygons in $\mathcal{F}$. Determine the largest possible sum of areas of all polygons in $\mathcal{F}$.