Solve the following system of equations:

$$
\begin{gathered}
x^{2}-y z=-23 \\
y^{2}-z x=-4 \\
z^{2}-x y=34
\end{gathered}
$$
From (2) subtract (1) and from (3) subtract (2), we get:

$$
(y-x)(x+y+z)=19
$$

$$
(z-y)(x+y+z)=38
$$

From these two equations, it follows that

$$
z=3 y-2 x
$$

Substituting this value of $z$ into (1) and (2):

$$
\begin{gathered}
x^{2}+2 x y-3 y^{2}=-23 \\
2 x^{2}-3 x y-y^{2}=-4
\end{gathered}
$$

After eliminating $x^{2}$, we obtain:

$$
y^{2}-x y=6
$$

from which

$$
x=\frac{y^{2}-6}{y}
$$

Substituting this value of $x$ into (6), we get:

$$
z=\frac{y^{2}+12}{y}
$$

Finally, substituting the values under (10) and (11) into (1), we get:

$$
y^{2}=36
$$

and thus

$$
y= \pm 6
$$

The values of $x$ and $z$ are obtained from the equations under (10) and (11):

$$
x= \pm 5, z= \pm 8
$$

(Emil Geist, real gymnasium VIII. class, Györ)

The problem was also solved by:

Mór Feuer, Bernát Friedmann, Károly Fröhlich, Zsigmond Goldstein, Béla Grünhut, Ervin Hofbauer, Nándor Kántor, Jenő Reif, István Szabó.