MathNet 2010 — Part 06 Question 48

Let $x$, $y$ and $z$ be non-zero real numbers such that $3x + 2y = z$ and $\frac{3}{x} + \frac{1}{y} = \frac{2}{z}$. Prove that $5x^2 - 4y^2 - z^2$ is always an integer.