, 2017

Solution:

Letting $I$ and $O$ denote the incenter and circumcenter of triangle $ABC$ we have by the triangle inequality that
$$AO \leq AI + OI \Longrightarrow R \leq \frac{r}{\sin \frac{A}{2}} + \sqrt{R(R-2r)}$$
and by plugging in our values for $r$ and $R$ we get
$$\sin \frac{A}{2} \leq \frac{17+\sqrt{51}}{34}$$
as desired. Equality holds when $ABC$ is isosceles and $I$ lies between $A$ and $O$.