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.
All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.
Prove that the midpoints of the sides of a regular polygon form a regular polygon.