RO-NMO 2000 Problem 113

A function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ is olympic if, any finite number of pairwise distinct elements of $\mathbb{R}^2$ at which the function takes the same value represent in the plane the vertices of a convex polygon.

Prove that if $p$ if a complex polynom of degree at least $1,$ then the function $\mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)|$ is olympic if and only if the roots of $p$ are all equal.