# 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.

×