Solve the following system of equations:

$$
\begin{gathered}
\frac{\sqrt{y}+5}{\sqrt{x}}+\frac{\sqrt{x}}{\sqrt{y}+5}=\frac{50}{7} \\
x+2 \sqrt{x y}+y=9
\end{gathered}
$$
$$
\begin{gathered}
\frac{\sqrt{y}+5}{\sqrt{x}}+\frac{\sqrt{x}}{\sqrt{y}+5}=\frac{50}{7} \\
x+2 \sqrt{x y}+y=9
\end{gathered}
$$

Let $\sqrt{x}=u$ and $\sqrt{y}=v$; then from (2) we get:

$$
u^{2}+2 u v+v^{2}=9
$$

from which

$$
u+v= \pm 3
$$

For $u+v=3$, from (1) we get:

$$
\frac{v+5}{3-v}+\frac{3-v}{v+5}=\frac{50}{7}
$$

Rearranging this equation, we obtain:

$$
v^{2}+2 v-8=0
$$

from which

$$
v_{1}=2, v_{2}=-4
$$

If $u+v=-3$, then from (1) we get:

$$
\frac{v+5}{3+v}+\frac{3+v}{v+5}=-\frac{50}{7}
$$

or

$$
16 v^{2}+128 v+247=0
$$

from which

$$
v_{3}=-\frac{13}{4}, v_{4}=-\frac{19}{4}
$$

The corresponding values of $u$ are obtained from (3):

$$
u_{1}=1, u_{2}=7, u_{3}=\frac{1}{4}, u_{4}=\frac{7}{4}
$$

Thus, the roots of the given equation are:

$$
x_{1}=1, y_{1}=4, x_{2}=49, y_{2}=16, x_{3}=\frac{1}{16}, y_{3}=\frac{169}{16}, x_{4}=\frac{49}{16}, y_{4}=\frac{361}{16}
$$

(Kármán Tódor.)

The problem was also solved by: Beck F., Bojedain F., Dénes A., Détshy K., Devecis M., Dolowschiák M., Fekete J., Freibauer E., Goldziher K., Kertész L., Kürth A., Laczkó E., Lukhaub Gy., Makk I., Petrogalli G., Porkoláb J., Probst E., Schiffer H., Schwartz E., Spitzer Ö., Spitzer S., Szabó I., Szabó K., Tüske J., Weisz Á., Weisz J.