Among the cylinders that can be inscribed in a right circular cone, determine the one whose lateral surface area is the largest.
Let the radius of the base of the cone be \( r \), the radius of the base of the cylinder \( r - x \), the height of the cone \( m \), and the height of the cylinder \( y \). The lateral surface area of the cylinder inscribed in the cone is:

\[
P = 2 (r - x) y \pi
\]

From similar triangles:

\[
x : y = r : m
\]

from which

\[
y = \frac{x m}{r}
\]

Substituting (2) into (1):

\[
P = 2 (r - x) \frac{x m}{r}
\]

The lateral surface area of the cone is maximized when

\[
z = -x^2 + r x
\]

takes its maximum value; \( z \) reaches its maximum when

\[
x = \frac{r}{2}
\]

Then from (2):

\[
y = \frac{m}{2}
\]

Thus, the radius of the base of the sought cylinder is half the radius of the base of the cone, and its height is half the height of the cone.

(Kántor Nándor.)

The problem was also solved by: Bauss O., Beck F., Dénes A., Devecis M., Fekete J., Friedmann B., Goldstein Zs., Goldziher K., Hofbauer E., Kármán T., Kertész L., Kornis Ö., Spitzer Ö., Szabó I., Szabó K., Thiringer A., Zerkovitz Zs.