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Functional equation from R^2 to R

Source: All-Russian Olympiad 2019 grade 10 problem 1

April 23, 2019
algebrafunctional equation

Problem Statement

Each point AA in the plane is assigned a real number f(A).f(A). It is known that f(M)=f(A)+f(B)+f(C),f(M)=f(A)+f(B)+f(C), whenever MM is the centroid of △ABC.\triangle ABC. Prove that f(A)=0f(A)=0 for all points A.A.
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