OMCC 2007 Problem 3

Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$ , $p$ and $q$ , there exist not necessarily distinct integers $a \neq 0$ , $b$ , $c$ belonging to $S$ , such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$ . Determine the maximum number of elements that $S$ can have.