In the isosceles triangle $ABC$, the base $BC$ is extended beyond $C$ to $D$ such that $BC = CD$. At point $D$, a perpendicular is erected to the line $BD$, and the plane is rotated around this perpendicular. The surface area and volume of the solid of revolution formed by the triangle $ABC$ need to be determined, given that the length of side $AB$ is $b$ and $\angle BAC = \alpha$.
The surface area of the solid of revolution is equal to the sum of the lateral surface areas of two frustums of a cone and the area of an annulus; for one of the frustums, the radii of the bases are $4 b \sin \frac{\alpha}{2}$ and $3 b \sin \frac{\alpha}{2}$, and for the other, they are $3 b \sin \frac{\alpha}{2}$ and $2 b \sin \frac{\alpha}{2}$; the slant height of both is (b); for the annulus, the radius of the larger circle is $4 b \sin \frac{\alpha}{2}$, and the smaller one is $2 b \sin \frac{\alpha}{2}$. Thus, the total surface area is

$$
\begin{gathered}
F=\left(4 b \sin \frac{\alpha}{2}+3 b \sin \frac{\alpha}{2}\right) \pi b+\left(3 b \sin \frac{\alpha}{2}+2 b \sin \frac{\alpha}{2}\right) \pi b+ \\
+\left(4 b \sin \frac{\alpha}{2}+2 b \sin \frac{\alpha}{2}\right) \pi 2 b \sin \frac{\alpha}{2}=12 \pi b^{2} \sin \frac{\alpha}{2}\left(1+\sin \frac{\alpha}{2}\right)
\end{gathered}
$$

The volume of the solid of revolution is equal to the difference in the volumes of the mentioned frustums of a cone

$$
\begin{gathered}
K=\frac{1}{3} \pi b \cos \frac{\alpha}{2}\left(16 b^{2} \sin ^{2} \frac{\alpha}{2}+12 b^{2} \sin ^{2} \frac{\alpha}{2}+9 b^{2} \sin ^{2} \frac{\alpha}{2}\right)- \\
-\frac{1}{3} \pi b \cos \frac{\alpha}{2}\left(9 b^{2} \sin ^{2} \frac{\alpha}{2}+6 b^{2} \sin ^{2} \frac{\alpha}{2}+\left(4 b^{2} \sin ^{2} \frac{\alpha}{2}\right)=\right. \\
=6 \pi b^{3} \sin ^{2} \frac{\alpha}{2} \cos \frac{\alpha}{2}
\end{gathered}
$$

(Jenö Filkorn, Nyitra.)

The problem was also solved by: Czank K., Faith F., Frank J., Gellért J., Keesz J., Kerekes T., Krausz B., Krisztián Gy., Kürth A., Russo M., Sasvári G., Scharff J., Smodics K., Stromfeld F., Gy. Lukhaub, Gy. Lupsa, G. Messik, Gy. Perl, H. Spitzer, E. Tézner, A. Weisz.