non-injective power mapping

Suppose $p$ is a prime and $f(a) = a^m$ is defined for $a$ that are relatively prime to $p$, then prove that $f$ is injective if and only if $m$ does not divide $p-1$.

Black and white painted sphere

The surface of a sphere is painted with black and white paints such that the area of black paint is less than 1/8 of the area of the sphere (the rest of the sphere is painted with white). Prove that one can inscribe a rectangular box in the sphere such that all eight corners land on white points.

More formally, if each point on the sphere is assigned a color black or white such that there is an injective but not surjective one-to-eight unordered mapping from the black points to the white points, prove that there are eight white points on the sphere that form a rectangular box.