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 19:04, 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.