Problem

Source: Romania TST 2023 Day 2 P4

Tags: geometry, combinatorics, combinatorial geometry



Fix a positive integer $n.{}$ Consider an $n{}$-point set $S{}$ in the plane. An eligible set is a non-empty set of the form $S\cap D,{}$ where $D$ is a closed disk in the plane. In terms of $n,$ determine the smallest possible number of eligible subsets $S{}$ may contain. Proposed by Cristi Săvescu