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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