1992 CMO 第 1 题

Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$ . Suppose that $\lambda$ ( $|\lambda|>1$ ) is a complex root of the equation, prove that $\lambda^{n+1}=1$ .

令方程 $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ 的实数系数满足 $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$ 。假设 $\lambda$ ( $|\lambda|>1$ ) 是方程的复根，证明 $\lambda^{n+1}=1$ 。