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 $m + n$ .