RO-NMO 2006 Problem 104

Let $n \in \mathbb N$ , $n \geq 2$ . Determine $n$ sets $A_i$ , $1 \leq i \leq n$ , from the plane, pairwise disjoint, such that:

(a) for every circle $\mathcal C$ from the plane and for every $i \in \left\{ 1,2,\ldots,n \right\}$ we have $A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi$ ;

(b) for all lines $d$ from the plane and every $i \in \left\{ 1,2,\ldots,n \right\}$ , the projection of $A_i$ on $d$ is not $d$ .