Originally posted by Dejection
Autumn 1998 (A Level) Tournament of the Towns
4. All diagonals of a regular 25-gon are drawn. Prove that no 0 of the diagonals pass through one interior point of the 25-gon.
Inscribe the 25-gon in a circle. The only line that could pass through the centre (if that is what you mean), must have a length of d as each vertex in a regular polygon touches the inscribing circle and with be perpendicular to the tangent at the intersection of the vertex and the circle. However, each diagonal must be less than d because the longest diagonals are not perpendicular to the tangent line (this is obvious by inspection).
Actually, the whole thing is pretty obvious by inspection.