\section*{Problem 1 - 291021}

Determine all pairs \((x ; y)\) of real numbers \(x\) and \(y\) that satisfy the following system of equations (1), (2):

\[
\begin{array}{r}
\frac{x}{y}=\frac{4}{9} \\
\frac{x+\sqrt{x}}{y+\sqrt{y}}=\frac{1}{2}
\end{array}
\]
}

If real numbers \(x\) and \(y\) satisfy equations (1) and (2), then:

From (1) we have \(y=\frac{9}{4} x(3)\), and from (2) we have \(2 x+2 \sqrt{x}=y+\sqrt{y}(4)\).

Substituting (3) into (4), we get

\[
2 x+2 \sqrt{x}=\frac{9}{4} x+\frac{3}{2} \sqrt{x} \quad \rightarrow \quad \frac{1}{4} x=\frac{1}{16} x^{2}
\]

From (1), \(x \neq 0\), so from (5) it follows that \(x=4\). From this and (3), we get \(y=9\).

Verification confirms that (1) and (2) are exactly satisfied by the pair \((x ; y)=(4 ; 9)\).