ILL 1985 Problem 57

a) The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$ , with edge length $1$ , as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$ b) Using the same assumptions in a), prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$