Difference between revisions of "1989 USAMO Problems/Problem 3"
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356637#356637 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356637#356637 Discussion on AoPS/MathLinks] | ||
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Revision as of 20:46, 3 July 2013
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.