Difference between revisions of "1989 USAMO Problems/Problem 1"
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− | == See | + | == See Also == |
{{USAMO box|year=1989|before=First question|num-a=2}} | {{USAMO box|year=1989|before=First question|num-a=2}} | ||
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356633#p356633 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356633#p356633 Discussion on AoPS/MathLinks] | ||
+ | {{MAA Notice}} | ||
[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 18:10, 18 July 2016
Problem
For each positive integer , let Find, with proof, integers such that and .
Solution
We note that for all integers ,
It then follows that
If we let , we see that is a suitable solution.
Notice that it is also possible to use induction to prove the equations relating and with .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1989 USAMO (Problems • Resources) | ||
Preceded by 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.