Question 184
Find the maximum value of the positive real number $k$ such that the inequality
$$
\frac{1}{kab + c^2} + \frac{1}{kbc + a^2} + \frac{1}{kca + b^2} \ge \frac{k+3}{a^2 + b^2 + c^2}
$$
holds for all positive real numbers $a, b, c$ such that $a^2 + b^2 + c^2 = 2(ab + bc + ca)$.
$$
\frac{1}{kab + c^2} + \frac{1}{kbc + a^2} + \frac{1}{kca + b^2} \ge \frac{k+3}{a^2 + b^2 + c^2}
$$
holds for all positive real numbers $a, b, c$ such that $a^2 + b^2 + c^2 = 2(ab + bc + ca)$.