5. A cube with side length 2 is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?
Solution:
$2 / 3$
First note that the long diagonal of the cube has length $2 \sqrt{3}$, so the radius of the sphere is $\sqrt{3}$. Let $x$ be the side length of the smaller cube. Then the distance from the center of the sphere to the far face of the smaller cube is $1+x$, while the distance from the center of the far face to a vertex lying on the sphere is $\frac{x \sqrt{2}}{2}$. Therefore, the square of the radius is $3=(1+x)^{2}+\frac{x^{2}}{2}$, or $3 x^{2}+4 x-4=(3 x-2)^{2}(x+2)=0$, so $x=\frac{2}{3}$.