Solve the following system of equations:

$$
\sqrt{5 \sqrt{x}+5 \sqrt{y}}+\sqrt{x}=10-\sqrt{y}
$$

$$
\sqrt{x^{3}}+\sqrt{y^{3}}=35
$$
Let $\sqrt{x}+\sqrt{y}=z$; then (1) can be written as:

$$
\sqrt{5 z}+z=10
$$

from which

$$
z_{1}=5, z_{2}=20
$$

If now $\sqrt{x}=u$ and $\sqrt{y}=v$, then

$$
\begin{gathered}
u+v=5 \text { and } u+v=20 \\
u^{3}+v^{3}=3 \text { and } u^{3}+v^{3}=35
\end{gathered}
$$

Solving these systems of equations and considering that $x=u^{2}$ and $y=v^{2}$, we get:

$$
x_{1}=y_{2}=4, x_{2}=y_{1}=9, x_{3}=y_{4}=\left(\frac{20-i \sqrt{131}}{2}\right)^{2}, x_{4}=y_{3}=\left(\frac{20+i \sqrt{131}}{2}\right)^{2}
$$

(Neidenbach Emil, Arad.)

The problem was also solved by: Ádámffy E., Bagyinka M., Baranyó A., Bartók I., Deutsch E., Deutsch J., Eckstein J., Enyedi B., Frank K., Haar A., Harsányi Z., Hirschfeld Gy., Jánosy Gy., Kertész G., Korény Gy., Kürti I., Liebner A., Ligeti P., Losonczy I., Messer P., Moskovits Zs., Pám M., Pivnyik I., Popoviciu M., Preisich G., Ragány B., Rássy P., Riesz K., Riesz M., Schlesinger O., Schwemmer I., Szücs A., Tóth B., Weber Gy., Weisz P.