What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular \(n\)-gon?
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.