Question 62
Fix a point $O$ in the plane and an integer $n \ge 3$. Consider a finite set $\mathcal{D}$ 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 $k \lt n$, 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 $\mathcal{D}$.
(a) No disc in $\mathcal{D}$ contains the point $O$; and
(b) For each positive integer $k \lt n$, 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 $\mathcal{D}$.