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 $i z_j$ . Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ 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 $\boxed{784}$ .

Solution 2

The equation $z^12-2^36$ can be factored as follows:

2011 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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2011 AIME II Problems/Problem 8

Contents

Problem

Solution

Solution 2

See also