Question 45
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
$$
2 S T \gt \sqrt{3(S + T)(S(b d + b f + d f) + T(a c + a e + c e))}
$$
$$
2 S T \gt \sqrt{3(S + T)(S(b d + b f + d f) + T(a c + a e + c e))}
$$