Balancing non-coplanar points - Colegiul Național din Iași (Iași - Roumanie)

Given n non-coplanar points (n ≥ 4), we will say we have balanced them if we find a plane equidistant from each of the n points.
a) Let A, B, C and D be four non-coplanar points. How many planes do these points balance?
b) Let A, B, C, D and E be five non-coplanar points. How many planes do these points balance?
In this article we proved that the solution provided in the book from which this problem was selected was wrong due to oversimplification and a mistake in the generalization of point a). Our solution took us through may areas of mathematics, such as space geometry, vectors and graphs, all combined into a creative and interesting proof.