Question 3
Find the largest positive real number $k$ such that
$$\frac1{kab+c^2}+\frac1{kbc+a^2}+\frac1{kca+b^2}\ge\frac{k+3}{a^2+b^2+c^2}$$
for every positive triple $(a,b,c)$ satisfying $a^2+b^2+c^2=2(ab+bc+ca)$.
$$\frac1{kab+c^2}+\frac1{kbc+a^2}+\frac1{kca+b^2}\ge\frac{k+3}{a^2+b^2+c^2}$$
for every positive triple $(a,b,c)$ satisfying $a^2+b^2+c^2=2(ab+bc+ca)$.