## Fall 2015, problem 6

Let $S$ denote the unit sphere in $\mathbb{R}^3$, let $x_1,\dots,x_n \in S$, and let $d(\cdot,\cdot)$ denote the standard distance function on $\mathbb{R}^3$. Prove that $$\sum_{i,j} d(x_i,x_j) \leq n^2$$ and determine for which points $x_1,\dots,x_n$ the above sum is $n^2$.