HMMT November, 2019

Solution:
By a smoothing argument, the largest triangle is that where the sides span $3$, $3$, and $2$ sides of the octagon respectively (i.e. it has angles $45^{\circ}$, $67.5^{\circ}$, and $67.5^{\circ}$), and the smallest triangle is that formed by three adjacent vertices of the octagon. Scaling so that the circumradius of the octagon is $1$, our answer is
$$\frac{\sin \left(90^{\circ}\right)+2 \sin \left(135^{\circ}\right)}{2 \sin \left(45^{\circ}\right)-\sin \left(90^{\circ}\right)}=\frac{1+\sqrt{2}}{\sqrt{2}-1}=3+2 \sqrt{2}$$
where the numerator is derived from splitting the large triangle by the circumradii, and the denominator is derived from adding the areas of the two triangles formed by the circumradii, then subtracting the area not in the small triangle.