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Unit Complex Numbers Absolute Value Optimization

2425MasterComplex NumbersOptimization

Harvard-MIT Mathematics Tournament

Let $x$ and $y$ be complex numbers such that $|x|=|y|=1$. a. Determine the maximum value of $|1+x|+|1+y|-|1+xy|$. b. Determine the maximum value of $|1+x|+|1+xy|+|1+xy^{2}|+\ldots+|1+xy^{2011}|-1006|1+y|$.
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