Difference between revisions of "1989 USAMO Problems/Problem 3"
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Since | Since | ||
<cmath> \lvert i- z_1 \rvert \cdot \lvert i - z_2 \rvert \dotsm \lvert i - z_n \rvert = \lvert P(i) \rvert < 1, </cmath> | <cmath> \lvert i- z_1 \rvert \cdot \lvert i - z_2 \rvert \dotsm \lvert i - z_n \rvert = \lvert P(i) \rvert < 1, </cmath> | ||
− | it follows that for some (not necessarily distinct) | + | it follows that for some (not necessarily distinct) conjugates <math>z_i</math> and <math>z_j</math>, |
<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 |
Revision as of 23:19, 14 April 2009
Problem
Let be a polynomial in the complex variable
, with real coefficients
. Suppose that
. Prove that there exist real numbers
and
such that
and
.
Solution
Let be the (not necessarily distinct) roots of
, so that
Since all the coefficients of
are real, it follows that if
is a root of
, then
, so
, the complex conjugate of
, is also a root of
.
Since
it follows that for some (not necessarily distinct) conjugates
and
,
Let
and
, for real
. We note that
Thus
Since
, these real numbers
satisfy the problem's conditions.
Resources
1989 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |