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

The twelve dots above represent the 12 roots of the equation $z^{12}-2^{36}=0$ . If we write $z=a+bi$ , then the real part of $z$ is $a$ and the real part of $iz$ is $-b$ . The blue dots represent those roots $z$ for which the real part of $z$ is greater than the real part of $iz$ , and the red dots represent those roots $z$ for which the real part of $iz$ is greater than the real part of $z$ . Now, the sum of the real parts of the blue dots is easily seen to be $8+16\cos\frac{\pi}{6}=8+8\sqrt{3}$ and the negative of the sum of the imaginary parts of the red dots is easily seen to also be $8+8\sqrt{3}$ . Hence our desired sum is $16+16\sqrt{3}=16+\sqrt{768}$ , giving the answer $\boxed{784}$ .

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2011 AIME II Problems/Problem 8

Problem

Solution