Suppose that \(d\) points are selected on the surface of a sphere.
Each point "induces" a hemisphere of the sphere. That is, it defines a hemisphere "centered" at that point (i.e. the "top" / "pole" of the hemisphere is at that point).
Prove that all \(d\) points are located within one same hemisphere \(\iff\) the \(d\) induced hemispheres all overlap somewhere.
Have fun!
