Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 4"
(→Solution) |
(category) |
||
Line 4: | Line 4: | ||
===Original statement=== | ===Original statement=== | ||
− | Let <math> | + | Let <math>n</math> be the smallest positive integer for which there exist positive real numbers <math>a</math> and <math>b</math> such that <math>(a+bi)^n=(a-bi)^n</math>. Compute <math>\frac{b^2}{a^2}</math>. |
==Solution== | ==Solution== | ||
Line 18: | Line 18: | ||
*[[Mock AIME 2 2006-2007]] | *[[Mock AIME 2 2006-2007]] | ||
− | [[Category:Intermediate | + | [[Category:Intermediate Algebra Problems]] |
Revision as of 21:20, 30 November 2007
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
.