In a certain right-angled triangle, the sum of the sides is $30 \mathrm{~cm}$, and the sum of the areas of the squares drawn over the sides is $338 \mathrm{~cm}^{2}$. What are the lengths of the sides?
If $x$ and $y$ are the legs and $z$ is the hypotenuse, then

$$
\begin{gathered}
x+y+z=30 \\
x^{2}+y^{2}+z^{2}=338
\end{gathered}
$$

and

$$
x^{2}+y^{2}=z^{2}
$$

Subtracting the third equation from the second, we get

$$
2 z^{2}=338 ; \quad z=13 \text{ cm}
$$

Substituting the value of $z$ into the first and second equations:

$$
x+y=17
$$

and

$$
x^{2}+y^{2}=169
$$

from which

$$
x_{1}=5 \text{ cm}, \quad y_{1}=12 \text{ cm} \quad \text { and } \quad x_{2}=12 \text{ cm}, \quad y_{2}=5 \text{ cm}
$$

(Gyula Kovács, Budapest.)

Number of solutions: 26.