HMMT February

Let $\angle A B D=\angle A D B=x$, and $\angle D A B=180-2 x$. In triangle $A C D$, by the law of sines, $C D=\frac{A D}{\sin \angle A C M} \cdot \sin 198-2 x$, and by the law of sines in triangle $B C D, C D=\frac{B D}{\sin \angle B C M} \cdot \sin x+36$. Combining the two, we have $2 \cos x=\frac{B D}{A D}=\frac{\sin 198-2 x}{\sin x+36} \cdot \frac{\sin \angle B C M}{\sin \angle A C M}$. But by the ratio lemma, $1=\frac{M B}{M A}=\frac{C B}{C A} \frac{\sin \angle B C M}{\sin \angle A C M}$, meaning that $\frac{\sin \angle B C M}{\sin \angle A C M}=\frac{C A}{C B}=\frac{\sin 36}{\sin 18}=2 \cos 18$. Plugging this in and simplifying, we have $2 \cos x=\frac{\sin 198-2 x}{\sin x+36} \cdot 2 \cos 18=\frac{\cos 108-2 x}{\cos 54-x} \cdot 2 \cos 18$, so that $\frac{\cos x}{\cos 18}=\frac{\cos 108-2 x}{\cos 54-x}$. We see that $x=36^{\circ}$ is a solution to this equation, and by carefully making rough sketches of both functions, we can convince ourselves that this is the only solution where $x$ is between 0 and 90 degrees. Therefore $\angle A B D=\angle A D B=36$, $\angle D A B=108$. Simple angle chasing yields $\angle A E B=72, \angle E C A=54, \angle E A C=54, \angle E A B=72$, making $D, A$, and $E$ collinear, and so $\angle B D E=36$. And because $A F=A B=A D, \angle F D B=1 / 2 \angle F A B=9$, so $\angle F D E=36-9=27$.