1988 CMO 第 1 题

Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$ , suppose that inequality

$$r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}$$

holds for arbitrary reals $x_1,x_2,\dots ,x_n$ . Find the values of $r_1,r_2,\dots ,r_n$ .

令 $r_1,r_2,\dots ,r_n$ 为实数。给定 $n$ 实数 $a_1,a_2,\dots ,a_n$ 不全部等于 $0$ ，假设不等式

$$r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}$$

对于任意实数 $x_1,x_2,\dots ,x_n$ 成立。查找 $r_1,r_2,\dots ,r_n$ 的值。