Problems

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 a convex quadrilateral $ABCD$ can be drawn inside a circle if and only if $\mathrm{\angle }ABC+\mathrm{\angle }CDA={180}^{\circ }$.

Prove that the midpoints of the sides of a regular polygon form a regular polygon.