# AIME 2015 Problem 15

Geometry Level 5

Circles $$\mathcal{P}$$ and $$\mathcal{Q}$$ have radii $$1$$ and $$4$$, respectively, and are externally tangent at point $$A$$. Point $$B$$ is on $$\mathcal{P}$$ and point $$C$$ is on $$\mathcal{Q}$$ so that line $$BC$$ is a common external tangent of the two circles. A line $$\ell$$ through $$A$$ intersects $$\mathcal{P}$$ again at $$D$$ and intersects $$\mathcal{Q}$$ again at $$E$$. Points $$B$$ and $$C$$ lie on the same side of $$\ell$$, and the areas of $$\triangle DBA$$ and $$\triangle ACE$$ are equal. This common area is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are relatively prime positive integers. Find $$m+n$$.

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