The lengths of the parallel sides of a certain trapezoid are $a$ and $b$, and its height is $m$. What is the length of the line that is parallel to $a$ and $b$ and divides the trapezoid into two parts whose areas are in the ratio $p: q$?
The sought line divides the trapezoid into two trapezoids, let the height of the lower one be $y$, and the height of the upper one $m-y$. Then, according to the problem,

$$
(a+x) y:(x+b)(m-y)=q: p
$$

furthermore

$$
\begin{gathered}
(a+x) y+(x+b)(m-y)=(a+b) m \\
y(a+x-x-b)=m(a+b-x-b) \\
y=\frac{m(a-x)}{a-b} \\
m-y=\frac{m(x-b)}{a-b}
\end{gathered}
$$

Substituting these values into 1), we get

$$
\begin{gathered}
p\left(a^{2}-x^{2}\right)=q\left(x^{2}-b^{2}\right) \\
(p+q) x^{2}=p a^{2}+q b^{2} \\
x=\sqrt{\frac{p a^{2}+q b^{2}}{p+q}}
\end{gathered}
$$

Sacher Aladár, fg. VIII. Losoncz.

The problem was also solved by: Bolemann Béla fg. VIII. Budapest; Friedmann Bernát, fg. VII. S. A. Ujhely; Grossmann Gusztáv, fg. VIII. Budapest; Jankovich György, fg. VIII. Losoncz; Meitner Elemér, fr. VIII. Budapest; Robitsek Ignácz, fr. VII. Györ; Suták Sándor, fg. VIII. Nyíregyháza; Visnya Aladár, fr. VII. Pécs; Weisz Lipót, fr. VI. Pécs.