Mock AIME II 2012 Problems/Problem 11
Revision as of 02:17, 5 April 2012 by Testingtesting (talk | contribs) (Created page with "==Problem== There exist real values of <math>a</math> and <math>b</math> such that <math>a+b=n</math>, <math>a^2+b^2=2n</math>, and <math>a^3+b^3=3n</math> for some value of <ma...")
Problem
There exist real values of and
such that
,
, and
for some value of
. Let
be the sum of all possible values of
. Find
.
Solution
First, if, then
. We now assume that
.
Now, note that .
Also, we have
.
Next, . But we know
, so
.
Since the only possible values of are
and
, our final answer is
.
(It is easy to check that there exists satisfying the equations.)