1989 USAMO Problems/Problem 3

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

Solution

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 $w$, is also a root of $P$.

Since \[\lvert i- z_1 \rvert \cdot \lvert i - z_2 \rvert  \dotsm \lvert i - z_n \rvert = \lvert P(i) \rvert < 1,\] it follows that for some (not necessarily distinct) conjugates $z_i$ and $z_j$, \[\lvert z_i-i \rvert \cdot \lvert z_j-i \rvert < 1.\] Let $z_i = a+bi$ and $z_j = a-bi$, for real $a,b$. We note that \[(a+b+1)^2 - (a+b-1)^2 = 4a + 4b .\] Thus \begin{align*} (a^2+b^2+1)^2 &= (a^2+b^2-1)^2 + 4a^2 + 4b^2 = \lvert a^2 + b^2 - 1 - 2ai \rvert ^2 + 4b^2 \\ &= \lvert (a-i)^2 - (bi)^2 \rvert^2 + 4b^2 \\ &= \bigl( \lvert a+bi - i \rvert \cdot \lvert a-bi -i \rvert \bigr)^2 + 4b^2 \\ &= \bigl( \lvert z_i - i \rvert \cdot \lvert z_j - i \rvert \bigr)^2 + 4b^2 < 1+4b^2. \end{align*} Since $P(a+bi) = P(z_i) = 0$, these real numbers $a,b$ satisfy the problem's conditions. $\blacksquare$

See Also

1989 USAMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5
All USAMO Problems and Solutions

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