MathDB

Beautiful combinatorial geometry

Source: BWM 2004, 2nd round, problem 2

September 1, 2004

geometryperimetergeometric transformationreflectionparameterizationfunctionintegration

Problem Statement

Let

k

be a positive integer. In a circle with radius

1

, finitely many chords are drawn. You know that every diameter of the circle intersects at most

k

of these chords. Prove that the sum of the lengths of all these chords is less than

k \cdot \pi