6. On the table, there are three cones standing on their bases, touching each other. The radii of their bases are 6, 24, and 24. A truncated cone is placed on the table with its smaller base down, and it shares a generatrix with each of the other cones. Find the radius of the smaller base of the truncated cone.
Answer: 2.
![](https://cdn.mathpix.com/cropped/2024_05_06_bd69057b1c245517edf6g-18.jpg?height=524&width=1312&top_left_y=714&top_left_x=428)

Solution. Let $C$ be the center of the smaller base of the truncated cone, and $r$ be the radius of this base. Let $O_{1}, O_{2}, O_{3}$ be the centers of the bases of the other cones. Denote by $\mathcal{K}_{0}$ the cone that completes the truncated cone to a regular cone, and by $\mathcal{K}_{1}$ the cone with the base center at $O_{1}$. On the left diagram, the section of $\mathcal{K}_{0}$ and $\mathcal{K}_{1}$ by the plane $\Pi$ passing through points $O_{1}$ and $C$ and perpendicular to the table is shown. By the problem's condition, $\mathcal{K}_{0}$ and $\mathcal{K}_{1}$ have a common generatrix, which lies in $\Pi$ since it passes through the vertices of the cones. Let $B$ be the point of intersection of this generatrix with the table. Then $B$ lies on the boundaries of the bases of $\mathcal{K}_{0}$ and $\mathcal{K}_{1}$, as well as on the segment $C O_{1}$ connecting the centers of the bases. Hence, the bases of $\mathcal{K}_{0}$ and $\mathcal{K}_{1}$ touch each other at point $B$, i.e., $B C = r$. Similarly, it can be verified that the distance from $C$ to the bases of the other two cones is also $r$. Therefore, the following equalities hold:

$$
r = C O_{1} - 6 = C O_{2} - 24 = C O_{3} - 24
$$

Then $C O_{2} = C O_{3}$, which means that point $C$ lies on the common tangent $A O_{1}$ to the bases of the larger cones (see the right diagram). Note that

$$
A O_{1} = \sqrt{O_{1} O_{2}^{2} - A O_{2}^{2}} = \sqrt{30^{2} - 24^{2}} = 18
$$

Therefore,

$$
(24 + r)^{2} = C O_{2}^{2} = 24^{2} + A C^{2} = 24^{2} + (A O_{1} - r - 6)^{2} = 24^{2} + (12 - r)^{2}, \quad \text{from which } r = 2
$$

## Variant 6