Task B-3.4. In the plane, 100 concentric circles are drawn with radii $1 \mathrm{~cm}, 2 \mathrm{~cm}, 3 \mathrm{~cm}, \ldots, 100 \mathrm{~cm}$. The circle with a radius of $1 \mathrm{~cm}$ is painted red, and all other areas bounded by consecutive circles are painted either red or green, but in such a way that adjacent areas are painted different colors. Determine the ratio of the area of all green regions to the area of all red regions.
## Solution.

The green areas are circular rings with a total area of

$$
Z=2^{2} \pi-1^{2} \pi+4^{2} \pi-3^{2} \pi+6^{6} \pi-5^{2} \pi+\ldots+100^{2} \pi-99^{2} \pi
$$

This can be written as

$$
\begin{aligned}
Z & =\pi((2-1)(2+1)+(4-3)(4+3)+(6-5)(6+5)+\ldots+(100-99)(100+99))= \\
& =\pi((2+1)+(4+3)+(6+5)+\ldots+(100+99))= \\
& =\pi(1+2+3+4+\ldots+99+100)= \\
& =\pi(1+100) \cdot 50=5050 \pi
\end{aligned}
$$

The area of the red regions is

$$
C=100^{2} \pi-Z=10000 \pi-5050 \pi=4950 \pi
$$

The desired ratio is

$$
\frac{Z}{C}=\frac{5050 \pi}{4950 \pi}=\frac{101}{99}
$$