Inequality of Numerators

Algebra Level 2

Let a,b,ca,b,c be positive reals. Also, let kk be the largest possible real such that

a1+b1+c1+a+b1+b+c1+c+a1a+b+ck.\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 kk can be expressed as pq\frac{p}{q} for relatively prime positive integers pp and qq, then what is p+q?p+q?


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