Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 12"
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+ | Let <math>A=2007</math> | ||
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+ | Solving: <math>(A-x)(x+A^{-k})^k=1</math> we note that the largest positive rational solution <math>x</math> is given by: | ||
+ | |||
+ | <math>x=A-\frac{1}{A^k}=\frac{A^{k+1}-1}{A^k}</math> | ||
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+ | ~Tomas Diaz. orders@tomasdiaz.com | ||
+ | |||
+ | {{alternate solutions}} |
Revision as of 21:10, 26 November 2023
Problem
Let be the largest positive rational solution
to the equation
for all integers
. For each
, let
, where
and
are relatively prime positive integers. If
what is the remainder when
is divided by
?
Solution
Let
Solving: we note that the largest positive rational solution
is given by:
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.