Maximizing in a Real Quadratic

Algebra Level 5

\( a, b\) and \(c\) are real values satisfying \( a + 2b + 3c = 195\) and \(a^2 + b^2 + c^2 = 2925 \). The maximum value of \(a\) has the form \( \frac {p}{q} \), where \(p\) and \(q\) are positive, coprime integers. What is the value of \( p + q\)?

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