8,9

Two circles touch each other internally. It is known that two radii of the larger circle, the angle between which is $60^{\circ}$, touch the smaller circle. Find the ratio of the radii of the circles.
Let circles with centers $O$ and $O_1$ and radii $R$ and $r (R > r)$ respectively touch internally at point $A$, and the radii $OB$ and $OC$ of the larger circle touch the smaller circle at points $M$ and $N$, respectively, with $\angle BOC = 60^\circ$.

Since the center of a circle inscribed in an angle lies on the bisector of this angle, $\angle AOB = 30^\circ$, and since the line of centers of two touching circles passes through their point of tangency,

$$
OO_1 = OA - O_1A = R - r
$$

From the right triangle $OO_1M$, we find that

$$
OO_1 = 2O_1M, \text{ or } R - r = 2r,
$$

from which $\frac{r}{R} = \frac{1}{3}$.

## Answer

$1: 3$