HMMT-Nov 2016 team, Problem 1

Two circles centered at $O_{1}$ and $O_{2}$ have radii 2 and 3 and are externally tangent at $P$. The common external tangent of the two circles intersects the line $O_{1} O_{2}$ at $Q$. What is the length of $P Q$ ?

Solution: Let the common external tangent intersect the circles centered at $O_{1}, O_{2}$ at $X, Y$ respectively. Then $\frac{O_{2} Q}{O_{1} Q}=\frac{O Y}{O X}=\frac{3}{2}$, so $\frac{O_{1} O_{2}}{O_{1} Q}=\frac{O_{2} Q-O_{1} Q}{O_{1} Q}=\frac{1}{2}$. Since $O_{1} O_{2}=2+3=5, O_{1} Q=10$ and hence $P Q=O_{1} Q+O_{1} P=12$. $\fbox{12}$.

Answer: 12