# Maximum With Two Constraints

Calculus Level 3

Let $$a$$, $$b$$ and $$c$$ be three real positive numbers such that $$a+b+c=12$$ and $$a^2+b^2+c^2=80$$. The maximum value of $$abc$$ is $$\frac{p\sqrt{q}}{r}$$, where $$p$$, $$q$$ and $$r$$ are positive integers, $$p$$ and $$r$$ are coprime and $$q$$ is square-free. Find $$p+q+r$$.

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