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Berkeley Math Circle Monthly Contest 4

Combinatorics Difficulty 5.0 AIME, harder Prove it United States

Problem:

2006 vertices of a regular 2007-gon are red. The remaining vertex is green. Let $G$ be the total number of polygons whose one vertex is green and the others are red. Denote by $R$ the number of polygons whose all vertices are red. Which number is bigger, $R$ or $G$ ? Explain your answer.

Solution

Solution:

We will prove that $G \geq R$ . For each polygon $\mathcal{P}$ with all red vertices we can correspond a polygon with one green vertex (namely we can add the green vertex to the set of vertices of $\mathcal{P}$ ). Thus $G \geq R$ . However, $G > R$ since the triangles with one green vertex can't be corresponded to some polygon whose all vertices are red.

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