Question 60
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 $2n$ (thus $|S| = 4n^2$). 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 $2n$ (thus $|S| = 4n^2$). 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 $\mathcal{F}$.
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 $2n$ (thus $|S| = 4n^2$). 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 $\mathcal{F}$.