In a right-angled triangle, the height dropped from the right angle to the hypotenuse forms angles with the bisectors of the acute angles, the ratio of which is $13: 17$. What are the measures of the acute angles of the triangle?
Let one of the acute angles of the right triangle be $\alpha$, the angle formed by the height and the two angle bisectors be $\varepsilon$ and $\eta$. Then

$$
\varepsilon=90^{\circ}-\frac{\alpha}{2} \text { and } 90^{\circ}-\frac{90^{\circ}-\alpha}{2}=45^{\circ}+\frac{\alpha}{2}
$$

According to the problem,

$$
\frac{\varepsilon}{\eta}=\frac{90^{\circ}-\frac{\alpha}{2}}{45^{\circ}+\frac{\alpha}{2}}=\frac{13}{17}
$$

from which

$$
\alpha=63^{\circ}
$$

(Vidor Andor, Györ.)