MathNet 2013 Question 90

Let $(a,b,c)$ be three distinct positive integers with $a+b+c=2013$. A step replaces $(x,y,z)$ by $(y+z-x,z+x-y,x+y-z)$. Prove that after 10 steps starting from $(a,b,c)$, the resulting triple contains at least one negative number.