<p>Let <img alt="$z=a+bi$" height="14" src="https://latex.artofproblemsolving.com/4/6/b/46b245331cc3cccf7ccaf7e2f6a34fa2131b8b33.png" width="79" /> be the complex number with <img alt="$\vert z \vert = 5$" height="18" src="https://latex.artofproblemsolving.com/b/a/d/bad98e37e80d20a26037f70b35ed48a0c82f7378.png" width="52" /> and <img alt="$b > 0$" height="13" src="https://latex.artofproblemsolving.com/0/f/f/0ff39ba262fdf0fc8425b3548976378ba294afba.png" width="40" /> such that the distance between <img alt="$(1+2i)z^3$" height="21" src="https://latex.artofproblemsolving.com/a/b/a/abaf67cd162bedf40f89a30e065df4a7e6fd33dc.png" width="79" /> and <img alt="$z^5$" height="15" src="https://latex.artofproblemsolving.com/f/7/7/f779e7d13031a8d827f9338bb8622a9f5e8c4dd1.png" width="15" /> is maximized, and let <img alt="$z^4 = c+di$" height="16" src="https://latex.artofproblemsolving.com/c/2/e/c2edff1ebd2d6c69ed66d83b6bac92d7e2bb1eb0.png" width="86" />. Find <img alt="$c+d$" height="14" src="https://latex.artofproblemsolving.com/d/d/5/dd54f8129186e28453c8c19f0c38ca25420a8e9f.png" width="39" />.</p>

Question No: 296

Let $z=a+bi$ be the complex number with $\vert z \vert = 5$ and $b > 0$ such that the distance between $(1+2i)z^3$ and $z^5$ is maximized, and let $z^4 = c+di$ . Find $c+d$ .

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Tags: Algebra Combinatorics Probability

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