Draw two circles with the same center, one with a radius of $1 \mathrm{~cm}$ and the other with a radius of $3 \mathrm{~cm}$. In the region outside the circle with a radius of $1 \mathrm{~cm}$ and inside the circle with a radius of $3 \mathrm{~cm}$, draw circles that are simultaneously tangent to both circles, as shown in the given figure.
(a) What should be the radius of these circles?

(b) What is the maximum number of these circles that can be drawn without overlapping?

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-031.jpg?height=252&width=257&top_left_y=248&top_left_x=1590)
(a) Since the circles with radii of 1 and $3 \mathrm{~cm}$ are concentric, the new circles tangent to the original ones must also have a radius of $1 \mathrm{~cm}$.

(b) The centers of the three circles with a radius of $1 \mathrm{~cm}$ shown in the figure form an equilateral triangle with a side length of 2 $\mathrm{cm}$. Therefore, their internal angles measure $60^{\circ}$. Since 360/60 = 6, we conclude that a maximum of six circles can be arranged without overlapping, under the given conditions.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-100.jpg?height=329&width=314&top_left_y=1072&top_left_x=1499)