A5. Inside a circle with radius 2, there are two equal circles with radius 1 that touch each other externally and touch the larger circle internally. Outside the two smaller circles and inside the larger circle, there is a smaller circle that touches the other three circles.

Calculate the radius of that smallest circle.

![](https://cdn.mathpix.com/cropped/2024_04_17_0b81ff2ef64511bfe8eeg-2.jpg?height=300&width=300&top_left_y=153&top_left_x=1523)
A5. The point of contact of the two circles with radius 1 is the center of the circle with radius 2. Let the center of the large circle be $M$, the centers of the circles with radius $1$ be $A$ and $B$, and the center of the smallest circle be $C$. Let the desired radius be $r$.

Then, in triangle $A M C$:

$A M=1, A C=1+r, M C=2-r$

Using the Pythagorean theorem, you find: $r=\frac{2}{3}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_f71f49936c57e5c73f63g-1.jpg?height=400&width=403&top_left_y=2104&top_left_x=1406)