IMC 1998 · Problem 5

Suppose that $\mathcal{S}$ is a family of spheres (i.e., surfaces of balls of positive radius) in $\mathbb{R}^n$, $n \ge 2$, such that the intersection of any two contains at most one point. Prove that the set $M$ of those points that belong to at least two different spheres from $\mathcal{S}$ is countable.