Problem

Source: Moldova TST 2021

Tags: function, algebra, geometry



Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.