Solve the following system of equations:

$$
\begin{gathered}
\frac{x+y}{1-x y}=1 \\
\frac{\left(1-x^{2}\right)\left(1-y^{2}\right)+4 x y}{\left(1+x^{2}\right)\left(1+y^{2}\right)}=\frac{1}{2}
\end{gathered}
$$
Solve the following system of equations:

$$
\begin{gathered}
\frac{x+y}{1-x y}=1 \\
\frac{\left(1-x^{2}\right)\left(1-y^{2}\right)+4 x y}{\left(1+x^{2}\right)\left(1+y^{2}\right)}=\frac{1}{2}
\end{gathered}
$$

The equation under 2) can also be written as:

$$
\frac{(1+x-y-x y)(1-x+y-x y)+4 x y}{1+x^{2}+y^{2}+x^{2} y^{2}}=\frac{1}{2}
$$

In the numerator of this, the value of $1-x y$ from 1) can be substituted, resulting in the following form:

$$
16 x y=1+x^{2}+y^{2}+x^{2} y^{2}
$$

If we now substitute the value of $x^{2}+y^{2}$ from 1) into this, we get the following equation:

$$
x^{2} y^{2}-10 x y+1=0
$$

From this,

$$
x_{1} y_{1}=5+2 \sqrt{6}, \quad x_{2} y_{2}=5-2 \sqrt{6}
$$

and substituting these values into 1) gives:

$$
x_{1}+y_{1}=-4-2 \sqrt{6}, \quad x_{2}+y_{2}=-4+2 \sqrt{6}
$$

From equations 5) and 6), it follows that $x_{1}$ and $y_{1}$, and respectively $x_{2}$ and $y_{2}$, are the roots of the following quadratic equations:

$$
\begin{aligned}
& z^{2}+(4-2 \sqrt{6}) z+(5+\sqrt{6})=0 \\
& u^{2}+(4-2 \sqrt{6}) u+(5-\sqrt{6})=0
\end{aligned}
$$

from which

$$
\begin{aligned}
& x_{1}=-2-\sqrt{6}+\sqrt{5+2 \sqrt{6}} \\
& y_{1}=-2-\sqrt{6}-\sqrt{5+2 \sqrt{6}} \\
& x_{2}=-2+\sqrt{6}+\sqrt{5-2 \sqrt{6}} \\
& y_{2}=-2+\sqrt{6}-\sqrt{5-2 \sqrt{6}}
\end{aligned}
$$

Note. $\sqrt{5+2 \sqrt{6}}=\sqrt{3}+\sqrt{2}$ and $\sqrt{5-2 \sqrt{6}}=\sqrt{3}-\sqrt{2}$, as can be verified by squaring. Thus, the values of $z$ and $u$ can also be written as:

$$
z= \pm(\sqrt{2}+\sqrt{3})(1 \mp \sqrt{2})
$$

and

$$
u= \pm(\sqrt{2}-\sqrt{3})(1 \mp \sqrt{2})
$$

Editor.

Gyozo Pronai, 8th grade student, Besztercebanya.

The problem was also solved by: Gustav Grossmann, 8th grade student, Budapest; Tivadar Heymann, 8th grade student, Györ; Elemér Meitner, 8th B grade student, V. district, Budapest; Lipót Weisz, 6th grade student, Pécs.