MathNet 2011 Question 61

The numbers $x$, $y$, $z$, $t$, $a$ and $b$ are positive integers, so that $xt - yz = 1$ and $\frac{x}{y} \gt \frac{a}{b} \gt \frac{z}{t}$. Prove that $ab \ge (x+z)(y+t)$.