HMMT November

The key claim is that $\mathbb{E}\left[X Y^{2}\right]=d^{2}+r_{1}^{2}+r_{2}^{2}$. To prove this claim, choose an arbitrary point $B$ on $\omega_{2}$. Let $r_{1}, r_{2}$ be the radii of $\omega_{1}, \omega_{2}$ respectively, and $O_{1}, O_{2}$ be the centers of $\omega_{1}, \omega_{2}$ respectively. Thus, by the law of cosines, $\overline{O_{1} B}=\sqrt{d^{2}+r_{2}^{2}-2 r_{2} d \cos (\theta)}$, where $\theta=\angle O_{1} O_{2} B$. Since the average value of $\cos (\theta)$ is 0 , the average value of ${\overline{O_{1} B}}^{2}$ is $d^{2}+r_{2}^{2}$. Now suppose $A$ is an arbitrary point on $\omega_{1}$. By the law of cosines, $\overline{A B}^{2}={\overline{O_{1} B}}^{2}+r_{1}^{2}-2 r_{1} d \cos (\theta)$, where $\theta=\angle A O_{1} B$. Thus, the expected value of $\overline{A B}^{2}$ is the expected value of ${\overline{O_{1} B}}^{2}+r_{1}^{2}$ which becomes $d^{2}+r_{1}^{2}+r_{2}^{2}$. This proves the key claim. Thus, we have $d^{2}+r_{1}^{2}+r_{2}^{2}=2023$. The lengths of the internal and the external tangents give us $d^{2}-\left(r_{1}+r_{2}\right)^{2}=361$, and $d^{2}-\left(r_{1}-r_{2}\right)^{2}=1369$. Thus, $$d^{2}-\left(r_{1}^{2}+r_{2}^{2}\right)=\frac{\left(d^{2}-\left(r_{1}+r_{2}\right)^{2}\right)+\left(d^{2}-\left(r_{1}-r_{2}\right)^{2}\right)}{2}=\frac{361+1369}{2}=865$$ Thus, $d^{2}=\frac{865+2023}{2}=1444 \Longrightarrow d=38$.