Inspired by Brandon Monsen

Geometry Level 5

A right triangle has legs aa and bb, and hypotenuse cc.

The length of the median to the hypotenuse is a3b+b3a3\sqrt[3]{a^{3}b+b^{3}a}.

If H\mathcal{H} is the minimum possible value of cc, find the value of 16H16 \mathcal{H}.


Inspiration.

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