3. In the sphere $T$ with radius $R$, eight spheres of equal radii $r$ are inscribed such that the centers of the inscribed spheres are the vertices of a cube, and each of the inscribed spheres touches the sphere $T$ and three of the inscribed spheres. Find the radius $r$ of the inscribed spheres.
Solution. If the sphere and the cube are intersected along the spatial diagonal of the cube, the resulting image will be as shown in the diagram. Here, $x$ represents half of the spatial diagonal of the cube, the edge of the cube is $2r$, and the spatial diagonal of the cube is equal to $2r\sqrt{3}$. From the diagram, it is clear that $R=r+x$, i.e., $R=r\sqrt{3}+r$, from which it follows that $r=\frac{R}{1+\sqrt{3}}$.

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