Inequality of Denominators

Algebra Level 5

Let a,b,ca,b,c be positive reals. Let kk be the largest possible real such that 1a+1b+1c+1a+b+1b+c+1c+aka+b+c\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 kk can be expressed as pq\dfrac{p}{q} for relatively prime positive integers p,qp,q, then find p+qp+q.


Inequality of Numerators, the easier version of this inequality.

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