INMO 2013 Problem 6

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