6. Three identical cones with vertex $A$ and vertex angle $\frac{\pi}{3}$ touch each other externally. Each of them touches internally a fourth cone with vertex at point $A$. Find the vertex angle of the fourth cone. (The vertex angle of a cone is the angle between its generators in the axial section.)
Answer: $\frac{\pi}{3}+2 \arcsin \frac{1}{\sqrt{3}}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9304f7a3bc6f36cb1f9ag-16.jpg?height=426&width=260&top_left_y=1386&top_left_x=961)

Solution. Let the desired angle be $2 \gamma$. Inscribed in the equal cones are spheres with centers $O_{1}, O_{2}, O_{3}$ of the same radius $r$. They obviously touch each other. Let the fourth cone touch these spheres at points $B, C, D$. Then $A B=A C=A D$ and, therefore, these points lie on a certain sphere with center $O$ and radius $R$, inscribed in the fourth cone. This sphere touches the spheres inscribed in the equal cones (for example, the spheres with centers at $O$ and $O_{1}$ touch at point $B$ in the plane containing the generatrix $A B$ and touching the fourth cone). Therefore, the points $O_{1}, O_{2}, O_{3}$ lie on the segments $O B, O C, O D$ respectively. Then $O O_{1}=O O_{2}=O O_{3}=R-r$. Moreover, the right triangles $A B O_{1}, A C O_{2}$, and $A D O_{3}$ are equal by two legs, so $A O_{1}=A O_{2}=A O_{3}$. This means that the line $A O$ passes through the center $H$ of the circumscribed circle of the triangle $O_{1} O_{2} O_{3}$ perpendicular to its plane. Note that this triangle is equilateral and $O_{1} O_{2}=2 r$, from which $O_{1} H=\frac{2 r}{\sqrt{3}}$. On the other hand,

$$
O_{1} H=O O_{1} \cdot \cos \angle O O_{1} H=(R-r) \cos \gamma
$$

Since $R=A B \cdot \operatorname{tg} \gamma=r \operatorname{ctg} \frac{\pi}{6} \operatorname{tg} \gamma=r \sqrt{3} \operatorname{tg} \gamma$, we get

$$
\frac{2 r}{\sqrt{3}}=r(\sqrt{3} \sin \gamma-\cos \gamma) \Longleftrightarrow \frac{1}{\sqrt{3}}=\sin \left(\gamma-\frac{\pi}{6}\right) \Longleftrightarrow 2 \gamma=\frac{\pi}{3}+2 \arcsin \frac{1}{\sqrt{3}}
$$

## Variant 5