2011 AIME II Problems/Problem 9
Problem 9
Let be non-negative real numbers such that
, and
. Let
and
be positive relatively prime integers such that
is the maximum possible value of
. Find
.
Solution
Note that none of the expressions involve products with
. The constraint is
, while the expression we want to maximize is
. Adding the left side of the constraint to the expression we get:
. This new expression is the product of three non-negative terms whose sum is equal to 1. By AM-GM this product is at most
. Since we have added at least
the desired maximum is at most
. It is easy to see that the maximum can in fact be achieved, so our answer is