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.
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.