2008 USAMO 第 5 题

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.

黑板上写着三个非负实数$r_1$、$r_2$、$r_3$。这些数具有以下性质：存在整数$a_1$、$a_2$、$a_3$，且不全为零，满足$a_1r_1 + a_2r_2 + a_3r_3 = 0$。我们被允许执行以下操作：在黑板上用$x\le y$找到两个数字$x$、$y$，然后擦除$y$并在其位置写上$y - x$。证明经过有限次数的此类操作后，我们最终可以在黑板上得到至少一个 $0$。