parmenides51 wrote: Let $a, b, c$ be positive real numbers such that $ab +bc + ca \le 3abc$. Prove that $$a^3+b^3+c^3 \ge a+b+c.$$ Let $a, b, c > 0$ satisfy $a + b + c \ge \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$. Prove that $$a^3 + b^3 + c^3 \ge a + b + c$$Let $a, b, c > 0$ satisfy $a b c =1$. Prove that $$a^3 + b^3 + c^3 \ge a + b + c$$