RMO Practice Problems - Algebra 4

Algebra Level 5

\[{\dfrac{a}{1+9bc + k(b-c)^{2}} + \dfrac{b}{1+9ca + k(c-a)^{2}} + \dfrac{c}{1+9ab + k(a-b)^{2}} \geq \dfrac{1}{2}}\]

Find the maximum value of real number \(k\) such that the above inequality holds for all non-negative real numbers \(a\), \(b\) and \(c\) satisfying \(a+b+c=1\).


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