Inequality of Denominators

Algebra Level 5

Let \(a,b,c\) be positive reals. Let \(k\) be the largest possible real such that \[\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}\ge \dfrac{k}{a+b+c}\]

If \(k\) can be expressed as \(\dfrac{p}{q}\) for relatively prime positive integers \(p,q\), then find \(p+q\).


Inequality of Numerators, the easier version of this inequality.

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