International Mathematical Olympiad 2002 — Official contest Question 6

Let $n \ge 3$ be a positive integer. Let $C_1$, $C_2$, …, $C_n$ be unit circles in the plane, with centers $O_1$, $O_2$, …, $O_n$ respectively. If no line meets more than two of the circles, prove that $$
\sum_{1 \le i \lt j \le n } \frac{1}{O_i O_j}
\le \frac{(n-1)\pi}{4}.
$$