USAMO 2008 Problem 5

Three nonnegative real numbers $r_1$ , $r_2$ , $r_3$ are written on a blackboard. These numbers have the property that there exist integers $a_1$ , $a_2$ , $a_3$ , not all zero, satisfying $a_1r_1 + a_2r_2 + a_3r_3 = 0$ . We are permitted to perform the following operation: find two numbers $x$ , $y$ on the blackboard with $x \le y$ , then erase $y$ and write $y - x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $0$ on the blackboard.