A pyramid has a base that is a right-angled quadrilateral, one side of which is $a=6 \mathrm{~cm}$, the other side is $b=8 \mathrm{~cm}$; each edge of the pyramid is $c=12 \mathrm{~cm}$. At what distance from the apex must the pyramid be cut by a plane parallel to the base so that the volumes of the resulting parts are equal?
Let the area of the quadrilateral formed by the section be $t$, the distance of the section from the vertex be $x$, and the height of the pyramid be $m$; according to the problem:

$$
\frac{a b m}{3}=\frac{2 t x}{3}
$$

$$
a b: t=m^{2}: x^{2}
$$

from which

$$
t=\frac{a b x^{2}}{m^{2}}
$$

Substituting (3) into (1), we get

$$
x=\frac{m}{\sqrt[3]{2}}
$$

If we denote the diagonal of the base by $d$, then:

$$
m=\sqrt{c^{2}-\frac{d^{2}}{4}}=\frac{1}{2} \sqrt{4 c^{2}-\left(a^{2}+b^{2}\right)}
$$

Substituting this value of $m$ into (4):

$$
x=\frac{\sqrt{4 c^{2}-\left(a^{2}+b^{2}\right)}}{2 \sqrt[3]{2}}=\frac{1}{4} \sqrt{4 c^{2}-\left(a^{2}+b^{2}\right)} \sqrt[3]{4}
$$

Substituting the given values, we get $x=8.658 \text{ cm}$.

(Frigyes Riesz, Gymn. VIII. class, Györ.)

The problem was also solved by: Béla Bálint, Mór Feuer, Antal Freund, Bernát Friedmann, Zsigmond Goldstein, Béla Grünhut, Ervin Hofbauer, Nándor Kántor, Béla Schneider, István Szabó.