31. Given an equilateral triangle, what is the ratio of the area of its circumscribed circle to the area of its inscribed circle?
31. Answer: 4.
Given an equilateral triangle $P Q R$. Let $C_{1}$ be its circumscribed circle and $C_{2}$ its inscribed circle. Suppose $Q R, R P, P Q$ are tangent to $C_{2}$ at $P^{\prime}, Q^{\prime}, R^{\prime}$, respectively. The area of triangle $P Q R$ is 4 times the area of triangle $P^{\prime} Q^{\prime} R^{\prime}$. So the area of $C_{1}$ is also 4 times the area of $C_{2}$.