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Minimum Ratio Complex Numbers First Quadrant

2175ExpertComplex NumbersInequalitiesOptimization

Harvard-MIT Mathematics Tournament

Let $a_{1}$, $a_{2}$, and $a_{3}$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of $$ \frac{\left|a_{1}+a_{2}+a_{3}\right|}{\sqrt[3]{\left|a_{1} a_{2} a_{3}\right|}} $$
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