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Let z be a complex number such that |z - 5 - i| = 5. Find the minimum value of $$|z - 1 + 2i|^2 + |z - 9 - 4i|^2$$.

Let z be a complex number such that z^5 = 1 and $$z \neq 1$$. Compute $$z + \frac{1}{z} + z^2 + \frac{1}{z^2}.$$

Apr 18, 2019

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For the first problem, the minimum value is 32.

For the second problem, the answer is 5.

Nov 29, 2019