RMO 2013 Questions, Solutions, Discussions
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(RMO 2013d P6, AoPS) Suppose that the vertices of a regular polygon of \(20\) sides are coloured with three colors red, blue and green, such that there are exactly three red vertices. Prove that there are three vertices \(A,B,C\) of the polygon having the same colour such that triangle \(ABC\) is isosceles.
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Decompose the set of vertices of the \(20\)-gon using the vertices of the pentagons as in the image below.
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Since there are exactly three red vertices and any of the remaining \(17\) vertices are blue or green, it follows that at least \(9\) of these \(17\) vertices are of the same color, say blue. Note that the set of vertices of a regular \(20\)-gon can be written as the union of the four pairwise disjoint sets, each of them consisting of the vertices of a regular pentagon (as in Fig. 1). Since there are nine blue vertices, by the pigeonhole principle, at least one of these four sets contains three blue points. Since any three points on a regular pentagon form the vertices of an isosceles triangle, the statement follows.
Lecture notes for Math Olympiad (IOQM, RMO, INMO)
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