6. The region is bounded by two cones with a common base, the height of which is twice less than the slant height. Three spheres are placed in the region, touching each other externally. Two spheres are identical and touch both cones, while the third touches the boundary of the region. What is the maximum ratio of the radius of the third sphere to the radius of the first?
Answer: $\frac{7-\sqrt{22}}{3}$.
![](https://cdn.mathpix.com/cropped/2024_05_06_66368caef28b3de8e403g-31.jpg?height=504&width=1140&top_left_y=1618&top_left_x=524)

Solution. Let $O_{1}, O_{2}, O_{3}$ be the centers of the spheres, $BO$ be the height of one of the cones, $R$ be the radius of the base of the cones, $r$ and $x$ be the radii of the first two and the third sphere, respectively, and $t=\frac{x}{r}$. On the left diagram, a part of the section of the region by the plane $BOO_{1}$ is shown, and on the right, a part of the section by the plane $BOO_{3}$. From the left diagram, it is clear that

$$
R=AO=AO_{1}+O_{1}O=r\left(\frac{1}{\cos \frac{\pi}{3}}+1\right)=3r
$$

Let $OD$ be the perpendicular to the generator $BC$ of the cone on the right diagram, and $\alpha$ be the angle between $OD$ and $OO_{3}$. Note that $OO_{3}$ is the height of the isosceles triangle $O_{1}O_{2}O_{3}$ with legs $r+x$ and base $2r$, so $OO_{3}=\sqrt{(x+r)^{2}-r^{2}}=\sqrt{x^{2}+2rx}$. Then

$$
R \cos \frac{\pi}{3}=OD=O_{3}E+OO_{3} \cdot \cos \alpha \Longleftrightarrow \frac{3}{2}r=x+\cos \alpha \sqrt{x^{2}+2rx} \Longleftrightarrow 2 \cos \alpha=\frac{3r-2x}{\sqrt{x^{2}+2rx}}=\frac{3-2t}{\sqrt{t^{2}+2t}}
$$

The right-hand side of the last equation decreases as $t$ increases. Therefore, the maximum value of $t$ is achieved when $\cos \alpha$ is minimal, i.e., when $\alpha=\frac{\pi}{3}$ (this corresponds to the case where the center of the third sphere lies on the common base of the cones). Considering the condition $t \leqslant \frac{3}{2}$, we get

$$
\sqrt{t^{2}+2t}=3-2t \Longleftrightarrow t^{2}+2t=4t^{2}-12t+9 \Longleftrightarrow 3t^{2}-14t+9=0 \Longleftrightarrow t=\frac{7-\sqrt{22}}{3}
$$