We create some segments in a regular \(n\)-gon by joining endpoints of the \(n\)-gon. What’s the maximum number of such segments while ensuring that no two segments are parallel? The segments are allowed to be sides of the \(n\)-gon - that is, joining adjacent vertices of the polygon.
Prove that the midpoints of the sides of a regular polygon form a regular polygon.
In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.