IMO Shortlist 2008 problem G5

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Dodao/la: arhiva
2. travnja 2012.
Let k and n be integers with 0\le k\le n - 2. Consider a set L of n lines in the plane such that no two of them are parallel and no three have a common point. Denote by I the set of intersections of lines in L. Let O be a point in the plane not lying on any line of L. A point X\in I is colored red if the open line segment OX intersects at most k lines in L. Prove that I contains at least \dfrac{1}{2}(k + 1)(k + 2) red points.

Proposed by Gerhard Woeginger, Netherlands
Izvor: Međunarodna matematička olimpijada, shortlist 2008