Use strong induction on $n$. Hint 2If a disc $\mathcal{D}$ intersects at most $10k-1$ discs, we can delete it and use induction hypothesis. A nice choice of $\mathcal{D}$ would be the minimal disc. Hint 3Use a common trick in real analysis (e.g. Vitali covering lemma): expand $\mathcal{D}=\mathcal{B}(O,r)$ to $\mathcal{D'}=\mathcal{B}(O,3r)$.

Prove that all circles that intersect $\mathcal{D}$ lie completely in $\mathcal{D'}$. Now consider areas yields that at most $9k-1$ circles intersect $\mathcal{D}$.