Question 12
Let $n \geq 3$ be a positive integer. Let $C_{1}, C_{2}, C_{3}, \ldots, C_{n}$ be unit circles in the plane, with centres $O_{1}, O_{2}, O_{3}, \ldots, O_{n}$ respectively. If no line meets more than two of the circles, prove that
$$ \sum_{1\leq i \lt j\leq n}\frac{1}{O_{i} O_{j}}\leq\frac{(n-1)\pi}{4}. $$
$$ \sum_{1\leq i \lt j\leq n}\frac{1}{O_{i} O_{j}}\leq\frac{(n-1)\pi}{4}. $$