$A B C$ is a right-angled triangle with the sum of the two legs being $s$; the height on the hypotenuse is $m$. Calculate the sides and angles of the triangle.
Let the legs be $a$ and $b$, the hypotenuse $c$; then:

$$
a+b=s
$$

or

$$
a^{2}+2 a b+b^{2}=s^{2}
$$

and

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

From these equations, we can determine the sides of the triangle. Substituting (3) and (2) into (1a):

$$
c^{2}+2 c m=s^{2}
$$

from which

$$
c=-m \pm \sqrt{m^{2}+s^{2}}
$$

the square root can only be positive because $c>0$.

From (2) and (1), it follows that the legs are the roots of the following equation:

$$
x^{2}-s x+c m=0
$$

Solving this equation, we get:

$$
a=\frac{s}{2}+\frac{\sqrt{s^{2}-4 c m}}{2} \text { and } b=\frac{s}{2}-\frac{\sqrt{s^{2}-4 c m}}{2}
$$

The angles are determined by the following equations.

$$
\sin \alpha=\frac{m}{b}, \sin \beta=\frac{m}{a}
$$

(Károly Goldziher, VII.o. t., Budapest, Gymnasium.)

The problem was also solved by: Mór Feuer, Bernát Friedmann, Károly Fröhlich, Emil Geist, Zsigmond Goldstein, Béla Grünhut, Ervin Hofbauer, Nándor Kántor, Jenő Reif, Frigyes Riesz, István Szabó, Lipót Weisz.