IMC 2004 · Problem 4

Suppose $n \ge 4$ and let $M$ be a finite set of $n$ points in $\mathbb{R}^3$, no four of which lie in a plane. Assume that the points can be coloured black or white so that any sphere which intersects $M$ in at least four points has the property that exactly half of the points in the intersection of $M$ and the sphere are white. Prove that all of the points in $M$ lie on one sphere.