59th Ukrainian National Mathematical Olympiad

Firstly, we will show that $n=5$ satisfies the conditions (Fig. 1).

Suppose there exists a configuration that satisfies the conditions with 4 rays (Fig. 2). Then these rays form 6 different angles. Thus, if conditions were satisfied then all the pairs of angles would be among these 6 angles. But if the greatest angle $\angle A_{1}O{A}_4$ equals $60^{\circ }$, then another $60^{\circ }$ angle will not exist. If $\angle A_{1}O{A}_4<60^{\circ }$, then there are no $60^{\circ }$ angles. If $\angle A_{1}O{A}_4>60^{\circ }$, then there won't be enough angles to satisfy all the conditions.