Difference between revisions of "1989 USAMO Problems/Problem 3"
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Let <math>z_1, \dotsc, z_n</math> be the (not necessarily distinct) roots of <math>P</math>, so that | Let <math>z_1, \dotsc, z_n</math> be the (not necessarily distinct) roots of <math>P</math>, so that | ||
<cmath> P(z) = \prod_{j=1}^n (z- z_j) . </cmath> | <cmath> P(z) = \prod_{j=1}^n (z- z_j) . </cmath> | ||
− | Since all the coefficients of <math>P</math> are real, it follows that if <math>w</math> is a root of <math>P</math>, then <math>P( \overline{w}) = \overline{ P(w)} = 0</math>, so <math>\overline{w}</math>, the [[complex conjugate]] of <math> | + | Since all the coefficients of <math>P</math> are real, it follows that if <math>w</math> is a root of <math>P</math>, then <math>P( \overline{w}) = \overline{ P(w)} = 0</math>, so <math>\overline{w}</math>, the [[complex conjugate]] of <math>w</math>, is also a root of <math>P</math>. |
Since | Since |
Revision as of 23:41, 25 April 2012
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 |