Difference between revisions of "Mock AIME I 2012 Problems/Problem 8"
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Revision as of 17:39, 7 April 2012
Problem
Suppose that the complex number satisfies . If is the maximum possible value of , can be expressed in the form . Find .
Solution
We begin by dividing both sides by to obtain . Now, consider that we may write with a positive real number so that and some real number. Then, Since we need , we must have , or equivalently, . By the quadratic equation, this has roots and to maximize , we take the larger root which is clearly maximized when is minimized. Since , the maximum value of will occur where , so the maximum value of occurs where and finally we find that the maximum value of is Taking the fourth power, the desired answer is .