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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