# Inspired by Brandon Monsen

Geometry Level 5

A right triangle has legs $$a$$ and $$b$$, and hypotenuse $$c$$.

The length of the median to the hypotenuse is $$\sqrt[3]{a^{3}b+b^{3}a}$$.

If $$\mathcal{H}$$ is the minimum possible value of $$c$$, find the value of $$16 \mathcal{H}$$.

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