CN-TST 2011 Problem 2

Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$ , and let $m_i$ be the multiplicity of $d_i$ , i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$ . Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$ .