RO-TST 1996 Problem 15

Let $S$ be a set of $n$ concentric circles in the plane. Prove that if a function $f: S\to S$ satisfies the property $$d( f(A),f(B)) \geq d(A,B)$$ for all $A,B \in S$ , then $d(f(A),f(B)) = d(A,B)$ , where $d$ is the euclidean distance function.