A beautifully constructed inequality---IIT 2006

Algebra Level 3

\(a,b,c\) are sides of a triangle \(ABC\) such that the quadratic equation (in \(x\)), \(x^2- 2x(a+b+c) + 3\lambda (ab + bc + ac) = 0 \) has real roots.

If the supremum of \(\lambda \) is \( \dfrac mn\), where \(m\) and \(n\) are coprime positive integers, submit your answer as \(\sqrt{m^2+ n^2} \).

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