In a right-angled triangle, the medians to the legs are given: $k_{a}=30 \mathrm{~cm}, k_{b}=40 \mathrm{~cm}$. What are the lengths of the sides and the angles of the triangle?
In the sense of the task:

$$
a^{2}+\frac{b^{2}}{4}=1600, b^{2}+\frac{a^{2}}{4}=900
$$

or

$$
4 a^{2}+b^{2}=6400,4 b^{2}+a^{2}=3600
$$

From these equations:

$$
a=\sqrt{\frac{4400}{3}}=20 \sqrt{\frac{11}{3}}, b=\sqrt{\frac{1600}{3}}=40 \sqrt{\frac{1}{3}}, c=\sqrt{2000}=20 \sqrt{5}
$$

Furthermore,

$$
\sin \alpha=\sqrt{\frac{11}{5}}, \sin \beta=\sqrt{\frac{4}{15}}
$$

from which

$$
\alpha=58^{\circ} 54^{\prime} 31^{\prime \prime}, \beta=31^{\circ} 5^{\prime} 29^{\prime \prime}
$$

(Mór Kelemen, Györ.)

The problem was also solved by: Braun J., Haar A., Ligeti P., Popoviciu M., Raab R., Söpkéz Gy.