SCHWEITZER 2001 Problem 1

Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$ . Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$ , then $f$ assumes at most $|S|$ distinct values.