Difference between revisions of "Mock AIME 3 2006-2007 Problems/Problem 4"
(Created page with "== Problem == Let <math>x=(i-11)(i-10)....(i+8)(i+9)</math> where <math>i=\sqrt{-1}</math>. If <math>a</math> and <math>b</math> are positive integers such that <math>x(a-bi)</ma...") |
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== Solution == | == Solution == | ||
+ | Pairing the terms <math>(i-k)</math> with <math>(i+k)</math> for <math>k=1,2,...9</math> all result in integers, so we're left with making <math>i(i-10)(i-11)</math> an integer. | ||
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+ | <math>i(i-10)(i-11)=(21+109i)</math>. To make this an integer, we just multiply it by its conjugate <math>(21-109i)</math>, so <math>a+b=21+109=\boxed{130}</math> |
Latest revision as of 23:35, 15 February 2015
Problem
Let where
. If
and
are positive integers such that
is an integer, then find the minimum value of
Solution
Pairing the terms with
for
all result in integers, so we're left with making
an integer.
. To make this an integer, we just multiply it by its conjugate
, so