1989 USAMO 第 3 题

Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.

设 $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ 为复数变量 $z$ 中的多项式，实数系数为 $c_k$。假设 $|P(i)| < 1 美元。证明存在实数$a$和$b$，使得$P(a + bi) = 0$且$(a^2 + b^2 + 1)^2 < 4 b^2 + 1$。