# Task 11.5. (12 points)

On the first step, a unit circle was drawn on a sheet of paper, and the circle bounded by it was colored black. On each subsequent step, for each of the circles drawn on the previous step, four new internally tangent circles of equal radii are drawn. These four circles are externally tangent to each other. The circles bounded by the new circles are colored white if the step number is even, or black if the step number is odd.

![](https://cdn.mathpix.com/cropped/2024_05_06_4f2747cb87be3ccfbc7fg-5.jpg?height=602&width=586&top_left_y=1738&top_left_x=1386)
The result of three steps is shown in the figure. The described process continues indefinitely. Find the area of the white region.
# Solution.

If at some step a circle of radius $R$ was drawn, then on the next step circles of radii $\frac{R}{1+\sqrt{2}}$ will be drawn. Thus, the area of the white region is the sum:

![](https://cdn.mathpix.com/cropped/2024_05_06_4f2747cb87be3ccfbc7fg-6.jpg?height=248&width=1258&top_left_y=470&top_left_x=450)

This is the sum of an infinite decreasing geometric progression with $b_{1}=\frac{4 \pi}{(1+\sqrt{2})^{2}}$ and

$q=-\frac{4}{(1+\sqrt{2})^{2}} . S=\frac{b_{1}}{1-q}=\frac{\frac{4 \pi}{(1+\sqrt{2})^{2}}}{1+\frac{4}{(1+\sqrt{2})^{2}}}=\frac{4 \pi}{(1+\sqrt{2})^{2}+4}$.

Answer: $S=\frac{4 \pi}{(1+\sqrt{2})^{2}+4}$.

| Criterion | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 12 |
| Correct answer. All main logical steps of the solution are provided. The solution contains arithmetic errors or typos that did not affect the overall course of the solution. | .+ | 10 |
| Correct answer. All main logical steps of the solution are provided. The solution lacks strict justification of some conclusions. | $\pm$ | 9 |
| The idea of the solution is found, but it is not completed or contains errors. However, a significant part of the solution is completed. | $+/ 2$ | 6 |
| The solution is generally incorrect or incomplete, but contains some progress in the right direction. | $\mp$ | 2 |
| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| Maximum score |  | 12 |