Solve the following system of equations:

$$
\begin{gathered}
x+y=6 \\
x^{4}+y^{4}=34 x y .
\end{gathered}
$$
Raising both sides of equation (1) to the fourth power and then subtracting (2), we get:

$$
4 x^{3} y+4 x y^{3}+6 x^{2} y^{2}+34 x y=6^{4}
$$

Adding $2 x^{2} y^{2}$ to both sides, we obtain:

$$
x y\left[4(x+y)^{2}+34\right]=6^{4}+2 x^{2} y^{2}
$$

or

$$
178 x y-2 x^{2} y^{2}=6^{4}
$$

or

$$
(x y)^{2}-89 x y+648=0
$$

From this equation, the values of $x y$ are: 81 and 8. Connecting these two values with (1), we get:

$$
x_{1}=y_{2}=3(1+2 i \sqrt{2}), x_{2}=y_{1}=3(1-2 i \sqrt{2}), x_{3}=y_{4}=4, x_{4}=y_{3}=2
$$

(Sándor Pichler, Budapest.)

The problem was also solved by: Ádámffy E., Bagyinka M., Baranyó A., Bartók I., Demjén E., Deutsch E., Deutsch I., Eckstein J., Enyedi B., Frank K., Freund E., Glück J., Haar A., Harsányi Z., Heimlich P., Hirschfeld Gy., Jánosy Gy., Kertész G., Kiss J., Korény Gy., Kürti I., Liebner A., Ligeti P., Losonczy I., Makó E., Messer P., Moskovits Zs., Neidenbach E., Pám M., Papp Gy., Pivnyik I., Popoviciu M., Preisich G., Ragány B., Rássy P., Riesz K., Riesz M., Sárközy E., Schlesinger O., Schuster Gy., Schwemmer I., Söpkéz Gy., Strobl I., Szőke D, Szücs A., Tóth B., Weber Gy., Weisz P.