2011 AIME II Problems/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 .
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