XXIV OM - I - Problem 10

Find the smallest natural number $ n > 1 $ with the following property: there exists a set $ Z $ consisting of $ n $ points in the plane such that every line $ AB $ ($ A, B \in Z $) is parallel to some other line $ CD $ ($ C, D \in Z $).
We will first prove that the set $ Z $ of vertices of a regular pentagon has the property given in the problem, that is, $ n \leq 5 $. We will show that each side of the regular pentagon is parallel to a certain diagonal and vice versa, each diagonal is parallel to a corresponding side.
It suffices to prove that $ AB \parallel CE $ (Fig. 11). Since a circle can be circumscribed around quadrilateral $ ABCE $ (namely, it is the circle circumscribed around the given regular pentagon), we have $ \measuredangle A + \measuredangle BCE = \pi $. Since $ \measuredangle A = \measuredangle B $, it follows that $ \measuredangle BCE = \pi - \measuredangle B $, which proves that $ AB \parallel CE $.
On the other hand, from the conditions of the problem, it follows that $ n \geq 4 $, since there are at least two different parallel lines, each containing at least two points of the set $ Z $. If $ n = 4 $ and points $ A, B, C, D $ satisfied the conditions of the problem, they would be the vertices of a trapezoid. However, none of the diagonals of a trapezoid is parallel to another line determined by its vertices. Therefore, $ n > 4 $, and from the previously proven inequality $ n \leq 5 $, it follows that $ n = 5 $.