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≤x,y,z≤n, show that the number of points in S is less than \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).Dan Schwarz, Romania analytic geometrymodular arithmeticcombinatorics unsolvedcombinatoricsnumber theorycombinatorial geometry