The lateral surface area of a certain regular octagonal pyramid is $2538.34 \mathrm{dm}^{2}$; the slant edges form an angle of $80^{\circ}$ with the base edges. What is the length of one base edge?

(Kövi Imre, teacher at the Igló fógymnasium).
Let the base edge of the pyramid be $x$; then the area of the base is: $2 x^{2} \cot 22.5^{\circ}$, and the lateral surface area: $2 x^{2} \cot 10^{\circ}$. According to the problem, therefore:

$$
2538.4=2 x^{2}\left(\cot 22^{\circ} 30^{\prime}+\cot 10^{\circ}\right)
$$

or

$$
2538.4=2 x^{2} \frac{\sin \left(22^{\circ} 30^{\prime}+10^{\circ}\right)}{\sin 22^{\circ} 30^{\prime} \sin 10^{\circ}}
$$

from which

$$
x=\sqrt{\frac{2538.4 \sin 22^{\circ} 30^{\prime} \sin 10^{\circ}}{2 \sin 32^{\circ} 30^{\prime}}}
$$

Performing the indicated operations, we get:

$$
x=12.53 \mathrm{dm}
$$

(István Szabó, forealisk. VII. o.t., Debrecen.)

The problem was also solved by:

Mór Feuer, Bernát Friedmann, Károly Fröhlich, János Galter, Emil Geist, Zsigmond Goldstein, Béla Grünhut, Ervin Hofbauer, Nándor Kántor, Béla Schneider.