CAPS 2009 Problem 6

Let $n\ge 16$ be an integer, and consider the set of $n^2$ points in the plane: $$G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}.$$ Let $A$ be a subset of $G$ with at least $4n\sqrt{n}$ elements. Prove that there are at least $n^2$ convex quadrilaterals whose vertices are in $A$ and all of whose diagonals pass through a fixed point.