HMMT February

By the Law of Sines we have $\sin \angle A=\frac{X Y}{A P}=\frac{4}{5}$. Let $I, T$, and $Q$ denote the center of $\omega$, the point of tangency between $\omega$ and $\Gamma$, and the center of $\Gamma$ respectively. Since we are told $A B C$ is acute, we can compute $\tan \angle \frac{A}{2}=\frac{1}{2}$. Since $\angle E A I=\frac{A}{2}$ and $\overline{A E}$ is tangent to $\omega$, we find $r=\frac{A E}{2}=\frac{15}{2}$. Let $H$ be the foot of the altitude from $A$ to $\overline{B C}$. Define $h_{T}$ to be the homothety about $T$ which sends $\Gamma$ to $\omega$. We have $h_{T}(\overline{A Q})=\overline{D I}$, and conclude that $A, T$, and $D$ are collinear. Now since $\overline{A P}$ is a diameter of $\Gamma, \angle P A T$ is right, implying that $D T H P$ is cyclic. Invoking Power of a Point twice, we have $225=A E^{2}=A T \cdot A D=A P \cdot A H$. Because we are given radius of $\Gamma$ we can find $A P=10$ and $A H=\frac{45}{2}=h_{a}$. If we write $a, b, c, s$ in the usual manner with respect to triangle $A B C$, we seek $B D \cdot D C=(s-b)(s-c)$. But recall that Heron's formula gives us $$\sqrt{s(s-a)(s-b)(s-c)}=K$$ where $K$ is the area of triangle $A B C$. Writing $K=r s$, we have $(s-b)(s-c)=\frac{r^{2} s}{s-a}$. Knowing $r=\frac{15}{2}$, we need only compute the ratio $\frac{s}{a}$. By writing $K=\frac{1}{2} a h_{a}=r s$, we find $\frac{s}{a}=\frac{h_{a}}{2 r}=\frac{3}{2}$. Now we compute our answer, $\frac{r^{2} s}{s-a}=\left(\frac{15}{2}\right)^{2} \cdot \frac{\frac{s}{a}}{\frac{s}{a}-1}=\frac{675}{4}$.