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Difference between revisions of "1989 USAMO Problems/Problem 3"

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<cmath> \lvert z_i-i \rvert \cdot \lvert z_j-i \rvert < 1. </cmath>
 
<cmath> \lvert z_i-i \rvert \cdot \lvert z_j-i \rvert < 1. </cmath>
 
Let <math>z_i = a+bi</math> and <math>z_j = a-bi</math>, for real <math>a,b</math>.  We note that
 
Let <math>z_i = a+bi</math> and <math>z_j = a-bi</math>, for real <math>a,b</math>.  We note that
−
<cmath> (a+b+1)^2 - (a+b-1)^2 = 4a^2 + 4b^2 . </cmath>
+
<cmath> (a+b+1)^2 - (a+b-1)^2 = 4a + 4b . </cmath>
 
Thus
 
Thus
 
<cmath> \begin{align*}
 
<cmath> \begin{align*}

Revision as of 22:22, 14 April 2009

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 $\overline{w}$, 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. $\blacksquare$

Resources

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