MathNet 2019-part-05 — 184 problems (EN–VI) Question 161

Each point $A$ in the plane is assigned a real number $f(A)$. It is known that $f(M) = f(A) + f(B) + f(C)$ whenever $M$ is the centroid of a triangle $ABC$. Prove that $f(A) = 0$ for every point $A$.