199 upvotes problem

Algebra Level 4

\[\large \dfrac{1}{a^{2}+9} + \dfrac{1}{b^{2}+9} + \dfrac{1}{c^{2}+9} \]

Given that \(a,b\) and \(c\) are positive reals with a sum of 1. If the maximum value of the above expression can be expressed as \(\dfrac{x}{y}\) where \(x\) and \(y\) are coprime integers, find the value of \(x+y\).

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