Difference between revisions of "1977 USAMO Problems/Problem 1"
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== See Also == | == See Also == | ||
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] | ||
Revision as of 18:04, 3 July 2013
Problem
Determine all pairs of positive integers 
such that
$(1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})$ (Error compiling LaTeX. Unknown error_msg) is divisible by $(1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})$ (Error compiling LaTeX. Unknown error_msg).
Solution
Denote the first and larger polynomial to be 
and the second one to be 
. In order for to be divisible by they must have the same roots. The roots of are the mth roots of unity, except for 1. When plugging into , the root of unity is a root of if the terms 
all represent a different mth root of unity.
Note that if 
, the numbers 
represent a complete set of residues modulo 
. Therefore, divides only if 
See Also
| 1977 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. 
