1989 USAMO Problems/Problem 3

Problem

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 $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$ .

Let $z_1, \dotsc, z_n$ be the (not necessarily distinct) roots of $P$ , so that $P(z) = \prod_{j=1}^n (z- z_j) .$ Since all the coefficients of $P$ are real, it follows that if $w$ is a root of $P$ , then $P( \overline{w}) = \overline{ P(w)} = 0$ , so $\overline{w}$ , the complex conjugate of $\overline{w}$ , is also a root of $P$ .

1989 USAMO (Problems)
Preceded by Problem 2	1 • 2 • 3 • 4 • 5	Followed by Problem 4
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