Spring 2018, problem 62

Suppose $n\geq 4$ and let $S$ be a finite set of points in $\mathbb{R}^3$, no four of which lie in a plane. Assume that the points in $S$ can be colored with red and blue such that any sphere which intersects $S$ in at least 4 points has the property that exactly half of the points in the intersection of $S$ and the sphere are blue. Prove that all the points of $S$ lie on a sphere.

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