RO-TST 1996 Problem 1

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function such that for every regular $n$ -gon $A_1A_2 \ldots A_n$ we have $f(A_1)+f(A_2)+\cdots +f(A_n)=0$ . Prove that $f(x)=0$ for all reals $x$ .