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n-gon function

Source: Romanian IMO Team Selection Test TST 1996, problem 1

September 13, 2005
functiongeometryrhombusgeometric transformationrotationperpendicular bisector

Problem Statement

Let f:R2→R f: \mathbb{R}^2 \rightarrow \mathbb{R} be a function such that for every regular n n -gon A1A2…An A_1A_2 \ldots A_n we have f(A1)+f(A2)+⋯+f(An)=0 f(A_1)+f(A_2)+\cdots +f(A_n)=0 . Prove that f(x)=0 f(x)=0 for all reals x x .
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