RO-TST 2013 Problem 12

Fix a point $O$ in the plane and an integer $n\geq 3$ . Consider a finite family $\mathcal{D}$ of closed unit discs in the plane such that: (a) No disc in $\mathcal{D}$ contains the point $O$ ; and (b) For each positive integer $k < n$ , the closed disc of radius $k + 1$ centred at $O$ contains the centres of at least $k$ discs in $\mathcal{D}$ . Show that some line through $O$ stabs at least $\frac{2}{\pi} \log \frac{n+1}{2}$ discs in $\mathcal{D}$ .