RMO Practice Problems - Algebra 4

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

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 $a+b+c=1$ .

Try this set RMO Practice Problems .