Mathematical Reflections 2014, Issue 2 - Problem J296

Problem:
Several positive integers are written on a board. At each step, we can pick any two
numbers $u$ and $v$, where $u \geq v$, and replace them with $u + v$ and $u - v$. Prove that after
a finite number of steps we can never obtain the initial set of numbers.

Proposed by Marius Cavachi.

Solution:
Let $S$ be the sum of the numbers on the blackboard at a given moment. After removing the numbers $u$ and $v$ and replacing them with $u+v$ and $u-v$ we have that the sum of the numbers on the blackboard is $$S'=S-u-v+(u+v)+(u-v)=S+(u-v)>S.$$ Therefore, after each step the sum of the numbers on the blackboard increases, and this proves that after a finite number of step we can never obtain the original numbers.