Inequality of Numerators

Algebra Level 2

Let \(a,b,c\) be positive reals. Also, let \(k\) be the largest possible real such that

\[\dfrac{a}{1}+\dfrac{b}{1}+\dfrac{c}{1}+\dfrac{a+b}{1}+\dfrac{b+c}{1}+\dfrac{c+a}{1}\le \dfrac{a+b+c}{k}.\]

If \(k\) can be expressed as \(\dfrac{p}{q}\) for relatively prime positive integers \(p\) and \(q\), then what is \(p+q?\)

Inequality of Denominators, the harder version of this inequality

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