Question 5
Let a,b,c be positive real numbers with abc=1. Prove
$$\begin{aligned}&(a^5+a^4+a^3+a^2+a+1)\\&\quad\cdot(b^5+b^4+b^3+b^2+b+1)\\&\quad\cdot(c^5+c^4+c^3+c^2+c+1)\\&\ge8(a^2+a+1)(b^2+b+1)(c^2+c+1).\end{aligned}$$
$$\begin{aligned}&(a^5+a^4+a^3+a^2+a+1)\\&\quad\cdot(b^5+b^4+b^3+b^2+b+1)\\&\quad\cdot(c^5+c^4+c^3+c^2+c+1)\\&\ge8(a^2+a+1)(b^2+b+1)(c^2+c+1).\end{aligned}$$