Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 12"
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+ | Let <math>A=2007</math> | ||
+ | |||
+ | 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_k=A-\frac{1}{A^k}=\frac{A^{k+1}-1}{A^k}=\frac{a_k}{b_k}</math> | ||
+ | |||
+ | Therefore <math>a_k=A^{k+1}-1</math>, and <math>b_k=A^k</math> | ||
+ | |||
+ | Then, <math>(A)(b_k)-a_k=(A)(A^k)-(A^{k+1}-1)=A^{k+1}-A^{k+1}+1=1</math> | ||
+ | |||
+ | <math>\sum_{k=2}^{A} ((A)(b_k)-a_k)=\sum_{k=2}^{A}1=A-1</math> | ||
+ | |||
+ | Therefore, <math>S=2007-1=2006 \equiv 6\;(mod\;1000)</math> | ||
+ | |||
+ | Reminder is <math>\boxed{6}</math> | ||
+ | |||
+ | ~Tomas Diaz. orders@tomasdiaz.com | ||
+ | |||
+ | {{alternate solutions}} |
Latest revision as of 21:17, 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:
Therefore , and
Then,
Therefore,
Reminder is
~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.