1. A circular coin $A$ is rolled, without sliding, along the circumference of another stationary circular coin $B$ with radius twice the radius of coin $A$. Let $x$ be the number of degrees that the coin $A$ makes around its centre until it first returns to its initial position. Find the value of $x$.
1. Answer. 1080
Solution. The number of complete revolutions the first coin $A$ has turned through is the sum of two components: the number of revolutions round the stationary coin $B$ if $A$ were sliding on $B$ and the number of revolutions round $A$ 's own axis (perpendicular to its plane and through its centre) determined by the distance travelled on the circumference of $B$. Thus, the total number of revolutions is
$$
1+\frac{2 \pi(2 r)}{2 \pi r}=3 .
$$

Hence the number of degrees $=3 \times 360=1080$.