Difference between revisions of "1974 USAMO Problems/Problem 1"
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{{USAMO box|year=1974|beforetext=|before=First Question|num-a=2}} | {{USAMO box|year=1974|beforetext=|before=First Question|num-a=2}} | ||
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=579731#579731 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=579731#579731 Discussion on AoPS/MathLinks] | ||
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Revision as of 18:55, 3 July 2013
Problem
Let ,
, and
denote three distinct integers, and let
denote a polynomial having all integral coefficients. Show that it is impossible that
,
, and
.
Solution
It suffices to show that if are integers such that
,
, and
, then
.
We note that
so the quanitities
must be equal in absolute value. In fact, two of them, say
and
, must be equal. Then
so
, and
, so
,
, and
are equal, as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1974 USAMO (Problems • Resources) | ||
First Question | Followed by Problem 2 | |
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.