## Task 10/82

Let the edge of a cube be the space diagonal of a cuboid with edge lengths $a, b$, and $c$. In what ratio do the edge lengths of the cuboid stand to each other if the volume of the cube is $3 \sqrt{3}$ times the volume of the cuboid?
The edge length of the cube is $d=\sqrt{a^{2}+b^{2}+c^{2}}$ for the volumes, this results in

$$
\begin{gathered}
\left(\sqrt{a^{2}+b^{2}+c^{2}}\right)^{3}=3 \sqrt{3 a b c}=\sqrt{27 a b c} \rightarrow\left(a^{2}+b^{2}+c^{2}\right)^{3}=3^{3} \cdot a^{2} b^{2} c^{2} \\
a^{2}+b^{2}+c^{2}=3 \sqrt[3]{a^{2} b^{2} c^{2}} \rightarrow \frac{a^{2}+b^{2}+c^{2}}{3}=\sqrt[3]{a^{2} b^{2} c^{2}}
\end{gathered}
$$

Thus, the arithmetic mean of the three non-negative numbers $a^{2}, b^{2}$, and $c^{2}$ is equal to their geometric mean. This is only the case when these numbers are all equal. Therefore, their non-negative square roots are also equal. Hence, $a=b=c$, and the cuboid is also a cube.