RMM 2009 Problem 2

A set $S$ of points in space satisfies the property that all pairwise distances between points in $S$ are distinct. Given that all points in $S$ have integer coordinates $(x,y,z)$ where $1 \leq x,y, z \leq n,$ show that the number of points in $S$ is less than $\min \Big((n + 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ Dan Schwarz, Romania