145. In a regular hexagonal pyramid, the center of the circumscribed sphere lies on the surface of the inscribed sphere. Find the ratio of the radii of the circumscribed and inscribed spheres.
145. There are two cases: 1) the center of the circumscribed sphere coincides with the center of the base, and 2) the center of the circumscribed sphere is located at a point on the surface of the inscribed sphere, diametrically opposite to the center of the base.

In the second case, denoting the radii of the inscribed and circumscribed spheres by $R$ and $r$, we find the height of the pyramid to be $2 r + R$ and the side of the base to be $\sqrt{R^{2} - 4 r^{2}}$. The section passing through the height and the midpoint of the side of the base of the pyramid is an isosceles triangle with height $R + 2 r$, base $\sqrt{3\left(R^{2} - 4 r^{2}\right)}$, and the radius of the inscribed circle equal to $r$. From this, we can derive the relationship for $R$ and $r$ as $3 R^{2} - 6 R r - 4 r^{2} = 0$.

Answer: $\frac{3+\sqrt{21}}{3}$ (in both cases).