Given two similar polygons, construct a third one that is similar to the previous ones and whose area is equal to the sum of the areas of the previous ones.
Let $a, b$ and $c$ be the corresponding sides of the given and the sought polygons, $t^{\prime}, t^{\prime \prime}$ and $T$ their areas. From the theorem of similarity, we know that:

$$
t^{\prime}: t^{\prime \prime}: T=a^{2}: b^{2}: c^{2}
$$

Furthermore, by assumption,

$$
T=t^{\prime}+t^{\prime \prime}
$$

If $t^{\prime}=k a^{2}$ according to (1), then $t^{\prime \prime}=k b^{2}$ and $T=k c^{2}$. Therefore, from (2) it follows that

$$
\begin{gathered}
k a^{2}+k b^{2}=k c^{2} \\
c^{2}=a^{2}+b^{2}
\end{gathered}
$$

which means that the sought side of the polygon can be constructed from the two corresponding sides based on the Pythagorean theorem. (Aladár Visnya, 7th grade student, Pécs).

The problem was also solved by: Tivadar Heymann and Sándor Suták.