Question 121
Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function satisfying the following property: If $A$, $B$, $C \in \mathbb{R}^2$ are the vertices of an equilateral triangle with sides of length $1$, then $f(A) + f(B) + f(C) = 0$. Prove that $f(x) = 0$ for all $x \in \mathbb{R}^2$.