IMO Shortlist 1983 problem 17

Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$ . Prove that $\max_{1 \leq i<j \leq n} P_iP_j > \frac{\sqrt 3}{2}(n -1) \min_{1 \leq i<j \leq n} P_iP_j$

Izvor: Međunarodna matematička olimpijada, shortlist 1983

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IMO Shortlist 1983 problem 17