2011 AIME II Problems/Problem 8

Problem:

Let $z_1$ , $z_2$ , $z_3$ , $\dots$ , $z_{12}$ be the 12 zeroes of the polynomial $z^{12} - 2^{36}$ . For each $j$ , let $w_j$ be one of $z_j$ or $iz_j$ . Then the maximum possible value of the real part of $\sum_{j = 1}^{12} w_j$ can be written as $m + \sqrt{n}$ , where $m$ and $n$ are positive integers. Find $m + n$ .

Solution: Let me first note that this is a hastily compiled solution.

<geogebra>973797c41824115b678d18138dff008b8acfda12</geogebra>

If you multiply one of those dots on the circle(of radius 8) by i, you move it 90 degrees clockwise. You want everything to be as far right as possible, and when you are done, you find that you get 784 when you add everything.

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2011 AIME II Problems/Problem 8