Symmetric Conditions But Not Symmetric Equality

Algebra Level 5

\[\large\sum_{\text{cyclic}(a,b,c)} \dfrac{a}{b+c}\]

Let \(a,b\) and \(c\) be positive reals satisfying \[ a^2+b^2+c^2=\dfrac{11}{7}(ab+bc+ca) \; . \]

If the sum of the minimum and maximum value of the expression above can be expressed as \(\dfrac{m}{n}\), where \(m\) and \(n\) are coprime positive integers, compute \(m+n\).

See my set of inequality problems: It's all about the inequality.

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