Solve the following system of equations:

$$
\begin{gathered}
x+y=4 z \\
x^{2}+y^{2}=17 z \\
x^{3}+y^{3}=76 z
\end{gathered}
$$
The second and third equations can also be written as:

$$
(x+y)^{2}=17 z+2 x y
$$

and

$$
(x+y)^{3}=76 z+3 x y(x+y)
$$

Substituting the value of $(x+y)$ from the first equation, we get:

$$
2 x y=16 z^{2}-17 z
$$

and

$$
3 x y=16 z^{2}-19
$$

thus

$$
3\left(16 z^{2}-17 z\right)=2\left(16 z^{2}-19\right)
$$

or

$$
16 z^{2}-51 z+38=0
$$

from which

$$
z_{1}=2 \text { and } z_{2}=\frac{19}{16}
$$

Substituting the values of $z_{1}$ and $z_{2}$ into the first and second equations, we get the following two systems of equations:

$$
\begin{array}{ll}
I . x+y=8 & I I \cdot x+y=\frac{19}{4} \\
x^{2}+y^{2}=34 & x^{2}+y^{2}=\frac{323}{16}
\end{array}
$$

From $I$:

$$
x_{1}=y_{2}=5 ; x_{2}=y_{1}=3
$$

From $I I$:

$$
x_{3}=y_{4}=\frac{19+\sqrt{285}}{8} \text { and } x_{4}=y_{3}=\frac{19-\sqrt{285}}{8} \text {. }
$$

(Jenő Rosenberg, Keszthely.)

The problem was also solved by: Ádámffy E., Bánó L., Dömény I., Erdős V., Égető B., Égető G., Fekete M., Friedländer H., Glück J., Haar A., Harsányi Z., Heimlich P., Hermann J., Jánosy Gy., Kertész G., Kiss J., Koffler B., Krampera Gy., Kürti I., Magyari F., Martini J., Messer P., Nagy A., Pám M., Rássy P., Riesz M., Sárközy P., Schöffer I., Schuster Gy., Schwarz Gy., Sonnenfeld J., Söpkéz Gy., Tandlich E., Tóth B., the math circle of the V. district high school.