$$
\begin{gathered}
\frac{2}{3 \sqrt{2 x+y}}-\frac{1}{2 \sqrt{2 x-y}}=\frac{1}{6} \\
15 \sqrt{2 x+y}+6 \sqrt{2 x-y}=8 \sqrt{4 x^{2}-y^{2}}
\end{gathered}
$$

The equations above need to be solved for $x$ and $y$.
By removing the denominators from the first equation:

$$
4 \sqrt{2 x-y}-3 \sqrt{2 x+y}=\sqrt{4 x^{2}-y^{2}}
$$

Multiplying (3) by 5 and adding to (2):

$$
26 \sqrt{2 x-y}=13 \sqrt{4 x^{2}-y^{2}}
$$

or

$$
2 \sqrt{2 x-y}=\sqrt{2 x-y} \sqrt{2 x+y}
$$

from which

$$
2 x+y=4
$$

Substituting (4) into (3):

$$
4 \sqrt{2 x-y}-6=2 \sqrt{2 x-y}
$$

from which

$$
2 x-y=9
$$

From (4) and (5):

$$
x=3.25, y=-2.5
$$

The problem was solved by: Bálint Béla, Fekete Jenő, Feuer Mór, Freund Antal, Friedmann Bernát, Friedman Gyula, Geist Emil, Goldstein Zsigmond, Grünhut Béla, Hofbauer Ervin, Hönigsfeld Tivadar, Jakobovits Dániel, Kántor Nándor, Klein Mór, Kornis Ödön, Piovarcsy Jenő, Riesz Frigyes, Schneider Béla, Simon Elek, Szita István, Weisz Ármin.