HMMT 2017 U16

Let $a$ and $b$ be complex numbers satisfying the two equations

$\begin{aligned} a^{3}-3 a b^{2} & =36 \\ b^{3}-3 b a^{2} & =28 i \end{aligned}$

Let $M$ be the maximum possible magnitude of $a$ . Find all $a$ such that $|a|=M$ .

Dataset text NuminaMath olympiads. Quoted with source.

Show the answer

3,-\frac{3}{2}+\frac{3 i \sqrt{3}}{2},-\frac{3}{2}-\frac{3 i \sqrt{3}}{2}

Solutions

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