USAMO 2001 Problem 5

Let $S$ be a set of integers (not necessarily positive) such that

(a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$ ; (b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$ .

Prove that $S$ is the set of all integers.