Question 93
In triangle $ABC$, $\angle A = 45^{\circ}$ and $M$ is the midpoint of $\overline{BC}$. $\overline{AM}$ intersects the circumcircle of $ABC$ for the second time at $D$, and $AM = 2MD$. Find $\cos \angle AOD$, where $O$ is the circumcenter of $ABC$.
