HMMT November, 2015

Solution:

Answer: $\frac{1+\sqrt{5}}{2}$

Let the shorter leg have length $\ell$, and the common ratio of the geometric sequence be $r>1$. Then the length of the other leg is $\ell r$, and the length of the hypotenuse is $\ell r^{2}$. Hence,
$$\begin{gathered} \ell^{2}+(\ell r)^{2}=\left(\ell r^{2}\right)^{2} \\ \Longrightarrow \ell^{2}\left(r^{2}+1\right)=\ell^{2} r^{4} \\ \Longrightarrow r^{2}+1=r^{4} \end{gathered}$$
Hence, $r^{4}-r^{2}-1=0$, and therefore $r^{2}=\frac{1 \pm \sqrt{5}}{2}$. As $r>1$, we have $r^{2}=\frac{1+\sqrt{5}}{2}$, completing the problem as the ratio of the hypotenuse to the shorter side is $\frac{\ell r^{2}}{\ell}=r^{2}$.