Question 8
Given six positive numbers $a, b, c, d, e, f$ such that $a\lt b\lt c\lt d\lt e\lt f$. Let $a+c+e=S$ and $b+d+f=T$. Prove that
$$
2 S T\gt \sqrt{3(S+T)(S(b d+b f+d f)+T(a c+a e+c e))}
$$
$$
2 S T\gt \sqrt{3(S+T)(S(b d+b f+d f)+T(a c+a e+c e))}
$$