A trip along Route 66

Algebra Level 5

\[ \large \begin{cases} {\displaystyle\sum_{k=1}^{66} a_{k} = 2015} \\ {\displaystyle\sum_{k=1}^{66} a_{k}^{2} = 62465} \end{cases} \]

Suppose real numbers \(a_1, a_2, a_3, \ldots,a_{66} \) satisfy the two equations above with \(S = \max(a_1, a_2, a_3, \ldots, a_{66}) \).

If \(S = \dfrac{m}{n},\) where \(m\) and \(n\) are positive coprime integers, then find \(m + n.\)

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