Mekkora az egyenló oldalú kúp, egyenló oldalú henger és gömb köbtartalmának aránya, ha felszíneik egyenlők?

What is the ratio of the volumes of an equilateral cone, an equilateral cylinder, and a sphere if their surface areas are equal?
Let the radii of the bases of the first two solids and the sphere be $r_{1}, r_{2}, r_{3}$, their volumes be $K_{1}, K_{2}, K_{3}$, and the common surface area be $F$.

The surface area of the equilateral cone

$$
\begin{gathered}
F=3 r_{1}^{2} \pi, \quad \text { from which } \\
r_{1}=\sqrt{\frac{F}{3 \pi}}
\end{gathered}
$$

The surface area of the equilateral cylinder

$$
\begin{gathered}
F=6 r_{2}^{2} \pi, \text { from which } \\
r_{2}=\sqrt{\frac{F}{6 \pi}}
\end{gathered}
$$

The surface area of the sphere

$$
\begin{gathered}
F=4 r_{3}^{2} \pi, \text { from which } \\
r_{3}=\sqrt{\frac{F}{4 \pi}}
\end{gathered}
$$

Using (1), (2), and (3), the corresponding volumes are:

$$
\begin{gathered}
K_{1}=\frac{r_{1}^{3} \pi \sqrt{3}}{3}=\frac{F \sqrt{F}}{3 \pi \sqrt{3 \pi}} \cdot \frac{\pi \sqrt{3}}{3}=\frac{F \sqrt{F}}{9 \sqrt{\pi}} \\
K_{2}=2 r_{2}^{3} \pi=\frac{F \sqrt{F}}{6 \pi \sqrt{6 \pi}} \cdot 2 \pi=\frac{F \sqrt{F} \sqrt{6}}{18 \sqrt{\pi}} \\
K_{3}=\frac{4}{3} r_{3}^{3} \pi=\frac{F \sqrt{F}}{4 \pi \sqrt{4 \pi}} \cdot \frac{4 \pi}{3}=\frac{F \sqrt{F}}{6 \sqrt{\pi}}
\end{gathered}
$$

The desired ratio is therefore

$$
K_{1}: K_{2}: K_{3}=\frac{1}{9}: \frac{\sqrt{6}}{18}: \frac{1}{6}=2: \sqrt{2 \cdot 3}: 3
$$

Géza Biczó (Bp. II., Rákóczi g. IV. o. t.)