Problem 1.2. Two half-lines have their common origin at point $O$. A circle $C_{1}$ is tangent to both half-lines, whose center is located at a distance $d_{1}$ from $O$, and whose radius is $r_{1}$. Subsequently, circles $C_{n}$ are constructed such that $C_{n}$ is tangent to the half-lines, externally tangent to $C_{n-1}$, and the distance from its center to $O, d_{n}$, is less than $d_{n-1}$, for $n>1$. Find the sum of the areas of the circles bounded by the circumferences $C_{n}$, for all $n$, in terms of $r_{1}$ and $d_{1}$.
Solution Problem 1.2 It is clear from the figure that, by Thales' Theorem, $\frac{r_{n}}{d_{n}}=\frac{r_{1}}{d_{1}}$ for all $n$. We will call

![](https://cdn.mathpix.com/cropped/2024_04_26_23b9750f7dc8395e0b8dg-2.jpg?height=540&width=1168&top_left_y=358&top_left_x=501)

this value $\alpha$. Additionally, it is given that:

$$
\begin{gathered}
\frac{r_{n}}{r_{n+1}}=\frac{d_{n}}{d_{n+1}}=\frac{d_{n+1}+r_{n+1}+r_{n}}{d_{n+1}}=1+\alpha+\frac{r_{n}}{d_{n+1}}= \\
1+\alpha+\frac{r_{n}}{r_{n+1}} \frac{r_{n+1}}{d_{n+1}}=1+\alpha+\frac{r_{n}}{r_{n+1}} \alpha
\end{gathered}
$$

Solving for $\frac{r_{n}}{r_{n+1}}$ gives $\frac{r_{n}}{r_{n+1}}=\frac{1+\alpha}{1-\alpha}$, which is constant, so the radii of the circles form a geometric progression with ratio

$$
r=\frac{1-\alpha}{1+\alpha}=\frac{1-r_{1} / d_{1}}{1+r_{1} / d_{1}}=\frac{d_{1}-r_{1}}{d_{1}+r_{1}}
$$

The desired sum of areas is

$$
S=\pi \sum_{n=1}^{\infty} r_{n}^{2}=\pi \frac{r_{1}^{2}}{1-\left(\frac{d_{1}-r_{1}}{d_{1}+r_{1}}\right)^{2}}=\frac{\pi}{4} \frac{r_{1}\left(d_{1}+r_{1}\right)^{2}}{d_{1}}
$$