MathNet 2013 Question 24

Let $a, b, c, x, y, z$ be positive real numbers such that $a + b + c = x + y + z$ and $a b c = x y z$. Further, suppose that $a \leq x \lt y \lt z \leq c$ and $a \lt b \lt c$. Prove that $a = x$, $b = y$ and $c = z$.