Solution. The body whose vertices are the centers of the spheres is composed of two congruent pyramids joined at their bases. The base of the pyramid is a square with side $2 R$ and height $\frac{a}{2}-R$, where $R$ is the radius of the spheres. Therefore, we have

$$
V=\frac{8 R^{2}}{3}\left(\frac{a}{2}-R\right)
$$

Thus, to find the volume of the body, we need to find the radius $R$ of the spheres.
![](https://cdn.mathpix.com/cropped/2024_06_05_969e4efbfe816ce80b6fg-4.jpg?height=274&width=864&top_left_y=350&top_left_x=440)

In the left drawing above, one of the six pyramids is represented, and in the right drawing, the section $A B C$ is shown, where $D$ is the center of the sphere. We have

$$
\begin{gathered}
P_{A C D}+P_{B C D}+P_{A B D}=P_{A B C}, \\
\frac{\frac{a \sqrt{2}}{2} R}{2}+\frac{\frac{a \sqrt{2}}{2} R}{2}+\frac{a R}{2}=\frac{a \cdot \frac{a}{2}}{2},
\end{gathered}
$$

from which we get $R=\frac{a}{2(\sqrt{2}+1)}$. Substituting into (1) we get $V=\frac{a^{3} \sqrt{2}}{3(\sqrt{2}+1)^{3}}$.

## Third Year
4. A cube with edge $a$ is given. The faces of the cube serve as bases for six pyramids whose common vertex is the center of the cube. A sphere is inscribed in each of these pyramids. Find the volume of the body whose vertices are the centers of the spheres.