Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 4"
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Revision as of 15:28, 3 April 2012
Problem
Revised statement
Let and
be positive real numbers and
a positive integer such that
, where
is as small as possible and
. Compute
.
Original statement
Let be the smallest positive integer for which there exist positive real numbers
and
such that
. Compute
.
Solution
Two complex numbers are equal if and only if their real parts and imaginary parts are equal. Thus if we have
so
, not a positive number. If
we have
so
so
or
, again violating the givens.
is equivalent to
and
, which are true if and only if
so either
or
. Thus
.