Difference between revisions of "1991 USAMO Problems/Problem 4"
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=344949#344949 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=344949#344949 Discussion on AoPS/MathLinks] | ||
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Revision as of 20:52, 3 July 2013
Problem
Let where
and
are positive integers. Prove that
.
[You may wish to analyze the ratio for real
and integer
.]
Solution
Let us assume without loss of generality that . We then note that
Similarly,
We note that equations and
imply that
. Then
, so
Multiplying this inequality by
, we have
It then follows that
Rearranging this inequality, we find that
, as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
Resources
1991 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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.