Question 132
For a fixed positive integer $k$, let $K$ be the set of all lattice points $(x, y)$ in the plane whose two coordinates $x$ and $y$ are both nonnegative integers less than $2k$. Thus $|K| = 4k^{2}$.
Now let $V$ be a set of $k^{2}$ non-degenerate quadrilaterals with the following properties:
i) The vertices of all these quadrilaterals are elements of $K$.
ii) Every point of $K$ is a vertex of exactly one of the quadrilaterals in $V$.
Determine the largest possible value of the sum of the areas of all $k^{2}$ quadrilaterals in $V$.
Now let $V$ be a set of $k^{2}$ non-degenerate quadrilaterals with the following properties:
i) The vertices of all these quadrilaterals are elements of $K$.
ii) Every point of $K$ is a vertex of exactly one of the quadrilaterals in $V$.
Determine the largest possible value of the sum of the areas of all $k^{2}$ quadrilaterals in $V$.