Originally Posted by

**Soroban** Hello, maria18!

I found something, but I too haven't solved it yet.

Rationalize: .$\displaystyle \frac{p}{q} \;=\;\frac{x-y\sqrt{2009}}{y-z\sqrt{2009}} \cdot \frac{y+z\sqrt{2009}}{y+z\sqrt{2009}} \;=\;$ .$\displaystyle \frac{xy + xz\sqrt{2009} - y^2\sqrt{2009} - 2009yz}{y^2-2009z^2} $

. . . . . . . . . $\displaystyle \frac{p}{q} \;=\;\frac{xy - 2009 + (xz - y^2)\sqrt{2009}}{y^2-2009z^2} $

Since $\displaystyle p$ is an integer: .$\displaystyle xz-y^2 \:=\:0 \quad\Rightarrow\quad xz \:=\:y^2$