Inspired by Hung Woei Neoh

Geometry Level 4

\[x^3-24x^2+180x-420=0\]

A \(\triangle ABC\) has side lengths \(a\), \(b\) and \(c\), which are also the roots of the equation above. If \(a\cos A+b\cos B+c\cos C=\dfrac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Find \(p+q+10\).


Inspiration; Try this.

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