IR-TST 2012 Problem 3

Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions: $i)$ There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them. $ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X}$ be on at least one of them.

Find maximum of $\mid S\mid$ .

Proposed by Erfan Salavati