# 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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