A lead $12 \mathrm{dm}$-diameter sphere is cut by two parallel planes in such a way that the diameter is divided into 3 equal parts by the cuts. From the two resulting spherical segments, two equal spheres are cast, and from the remaining middle piece, a cylindrical pipe is made, whose inner diameter is the same as that of the spheres and whose thickness is $2 \mathrm{~cm} ;$ what are the diameters of the resulting spheres and how long is the pipe?
Let's denote the radius of the sphere as $R$ and the radius of the cut circles as $r$. Then,

$$
\begin{gathered}
r^{2}=\frac{2}{3} R \cdot \frac{3}{4} R=\frac{8}{9} R^{2}, \\
r=\frac{2}{3} R \sqrt{2}
\end{gathered}
$$

The volume of the resulting spherical segment is

$$
\begin{gathered}
V=\frac{1}{3} \pi \frac{4}{9} R^{2}\left(3 R-\frac{2}{3} R\right) \\
V=\frac{4}{27} \pi R^{2} \frac{7 R}{3}=\frac{28}{81} \pi R^{3} \\
V=\frac{28}{81} \pi \cdot 12^{3} \text{ dm}^{3}=\frac{28.64}{3} \pi \text{ dm}^{3}
\end{gathered}
$$

The radius of the sphere $\zeta$ is derived from the following equation:

$$
\begin{gathered}
\frac{4}{3} \zeta^{3} \pi=\frac{28.64}{3} \pi \text{ dm}^{3} \\
\zeta^{3}=28.16 \text{ dm}^{3} \\
\zeta=\sqrt[3]{448} \text{ dm}=7.6 \text{ dm}
\end{gathered}
$$

The volume of the spherical segment cut by parallel planes is

$$
\begin{gathered}
V^{\prime}=\frac{4}{3} R^{3} \pi-\frac{56}{81} R^{3} \pi \\
V^{\prime}=\left(\frac{108}{81}-\frac{56}{81}\right) R^{3} \pi \\
V^{\prime}=\frac{56}{81} R^{3} \pi
\end{gathered}
$$

The volume of the cylindrical pipe, with its unknown length denoted as $h$, is:

$$
\begin{gathered}
V^{\prime}=h\left\{(\zeta+0.2)^{2}-\zeta^{2}\right\} \pi \\
V^{\prime}=h\{(0.4 \zeta+0.04)\} \pi
\end{gathered}
$$

From this,

$$
\begin{aligned}
h & =\frac{V^{\prime}}{(0.4 \zeta+0.04) \pi} \\
h & =\frac{\frac{52}{81} R^{3} \pi}{(0.4 \zeta+0.04) \pi} \\
h & =\frac{52 R^{3}}{81(0.4 \zeta+0.04)} \\
h & =\frac{52.12^{3}}{81(0.4 \zeta+0.04)} \\
h & =\frac{52.64}{0.12+1.2 \zeta} \\
h & =\frac{52.16}{0.03+0.3 \zeta} \\
h & =360.2 \text{ dm}
\end{aligned}
$$

(Mór Eckstein, 8th grade student, Nyitra).