Question 150
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 $a_1 r_1 + a_2 r_2 + a_3 r_3 = 0$. 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.
(This problem was suggested by Kiran Kedlaya.)
(This problem was suggested by Kiran Kedlaya.)