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Geometry Level 5

x324x2+180x420=0x^3-24x^2+180x-420=0

A ABC\triangle ABC has sides a,ba,b and cc. It is known that a,ba,b and cc are also the roots of the equation above.

Given that cosA+cosB+cosC=pq\cos A+\cos B+\cos C =\dfrac{p}{q}, where pp and qq are coprime positive integers, find the value of 11(p+q)+2511(p+q)+25.


Inspiration.

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