Nemotron-IMO-Bench 2026 A45

Let $n>3$ be an integer and let $M$ be a set of $n$ points in the plane, no three being collinear. Suppose that there is no circle that contains all points of $M$. Find all functions $f: M\to \mathbf{R}$ such that for any circle $C$ containing at least three points of $M$ we have $$\sum_{P\in C\cap M} f(P)=0.$$