2. $[\mathbf{3}]$ Let $A B C$ be an equilateral triangle. Let $\Omega$ be its incircle (circle inscribed in the triangle) and let $\omega$ be a circle tangent externally to $\Omega$ as well as to sides $A B$ and $A C$. Determine the ratio of the radius of $\Omega$ to the radius of $\omega$.
Answer: 3 Label the diagram as shown below, where $\Omega$ and $\omega$ also denote the center of the corresponding circles. Note that $A M$ is a median and $\Omega$ is the centroid of the equilateral triangle. So $A M=3 M \Omega$. Since $M \Omega=N \Omega$, it follows that $A M / A N=3$, and triangle $A B C$ is the image of triangle $A B^{\prime} C^{\prime}$ after a scaling by a factor of 3 , and so the two incircles must also be related by a scale factor of 3 .