RO-TST 1996 Problem 12

Let $n\geq 3$ be an integer and let $p\geq 2n-3$ be a prime number. For a set $M$ of $n$ points in the plane, no 3 collinear, let $f: M\to \{0,1,\ldots, p-1\}$ be a function such that

(i) exactly one point of $M$ maps to 0, (ii) if a circle $\mathcal{C}$ passes through 3 distinct points of $A,B,C\in M$ then $\sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p$ .

Prove that all the points in $M$ lie on a circle.