Question 74
Fix two integers $m \geq 2$ and $n \geq 2$. Let $S$ be a set of lattice points lying in the Cartesian rectangle $[1, m] \times [1, n]$; a lattice point is a point both of whose coordinates are integers. Show that if $|S| \geq m+n+\left\lfloor \frac{1}{4} m+\frac{1}{4} n-\frac{1}{2} \right\rfloor$, then there exists a circle passing through at least four pairwise distinct points of $S$.