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 $x+y$ .