, 2013

Solution:

Answer: $\sqrt{\sqrt{39}}$ As a preliminary, we may compute that by the law of cosines, the ratio $\frac{A D}{B D} = \frac{3}{\sqrt{5}}$. Now, construct the point $P$ in triangle $A B D$ such that $\triangle A P B \sim \triangle A E D$. Observe that $\frac{A P}{A D} = \frac{A E \cdot A B}{A D \cdot A D} = \frac{B C}{B D}$ (where we have used first the similarity and then condition 3). Furthermore, $\angle C B D = \angle D A B - \angle D A E = \angle D A B - \angle P A B = \angle P A D$ so by SAS, we have that $\triangle C B D \sim \triangle P A D$.

Therefore, by the similar triangles, we may compute $P B = D E \cdot \frac{A B}{A D} = 5$ and $P D = C D \cdot \frac{A D}{B D} = 7$. Furthermore, $\angle B P D = 360 - \angle B P A - \angle D P A = 360 - \angle B C D - \angle D E A = 60$ and therefore, by the law of cosines, we have that $B D = \sqrt{39}$.