4. We will call a point convenient for a circle if the angle between the tangents drawn from this point to the circle is $60^{\circ}$. Two circles with centers $A$ and $B$ touch each other, and point $M$ is convenient for each of them. Find the ratio of the radii of the circles if $\triangle A B M$ is a right triangle.
Solution. Note (see Fig. 1) that a point $M$ is convenient for a circle with center $O$ if and only if $O M$ is twice the radius.

![](https://cdn.mathpix.com/cropped/2024_05_06_0339a10e8d70fdaba306g-2.jpg?height=343&width=562&top_left_y=454&top_left_x=427)

1

![](https://cdn.mathpix.com/cropped/2024_05_06_0339a10e8d70fdaba306g-2.jpg?height=308&width=294&top_left_y=474&top_left_x=1144)

$2 a$

![](https://cdn.mathpix.com/cropped/2024_05_06_0339a10e8d70fdaba306g-2.jpg?height=239&width=185&top_left_y=483&top_left_x=1501)

$2 b$

Without loss of generality, let the circles have radii $r_{A} \geqslant r_{B}$. Since they are tangent, $A B=r_{A} \pm r_{B}$ (plus for external tangency and minus for internal tangency, see Fig. 2). On the other hand, $A M=2 r_{A}$ and $B M=2 r_{B}$, because $M$ is convenient for both circles. In triangle $A M B$, side $A M$ is the hypotenuse, since $2 r_{A} \geqslant 2 r_{B}$ and $2 r_{A} \geqslant r_{A} \pm r_{B}$. We obtain the equation

$$
4 r_{A}^{2}=4 r_{B}^{2}+r_{A}^{2} \pm 2 r_{A} r_{B}+r_{B}^{2}
$$

that is, $3 r_{A}^{2}=5 r_{B}^{2} \pm 2 r_{A} r_{B}$. If the tangency is internal, then $r_{A}=r_{B}$, but then $A=B$ and $A M B$ will not be a triangle. If the tangency is external, then $r_{A}=\frac{5}{3} r_{B}$. Therefore, the ratio of the radii is $3: 5$.

Criteria. If it is not taken into account that the circles can touch internally, 2 points are deducted.