Prove the following inequality using am gm inequality

Question

Problem:

If $abc \le a+b+c$ then prove that $a^2+b^2+c^2\ge √3abc$

I squared the inequality to get $a^2+b^2+c^2+2ab+2bc+2ac\ge a^2b^2c^2$. Now I tried to apply am gm but it didn't work.I have to someway end up with $3a^2b^2c^2$ on rhs and then it will be easy.Any help

Answer 1

If $abc\leq0$, so it's obvuios.

Let $abc>0$.

Thus, $$\frac{a+b+c}{abc}\geq1$$ and we need to prove that: $$(a^2+b^2+c^2)^2\geq3a^2b^2c^2,$$ for which it's enough to prove that: $$(a^2+b^2+c^2)^2\geq3a^2b^2c^2\cdot\frac{a+b+c}{abc}$$ or $$\sum_{cyc}(a^4+2a^2b^2-3a^2bc)\geq0$$ or $$\sum_{cyc}(2a^4-2a^2b^2+6a^2b^2-6a^2bc)\geq0$$ or $$\sum_{cyc}(a^4-2a^2b^2+b^4+3a^2c^2-6c^2ab+3b^2c^2)\geq0$$ or $$\sum_{cyc}(a-b)^2((a+b)^2+3c^2)\geq0$$ and we are done!

For positive variables a proof a bit of shorter.

By AM-GM we obtain: $$a^2+b^2+c^2=\sqrt{(a^2+b^2+c^2)^2}\geq\sqrt{3(a^2b^2+a^2c^2+b^2c^2)}=$$ $$=\sqrt{\frac{3}{2}\sum_{cyc}(a^2b^2+a^2c^2)}\geq \sqrt{\frac{3}{2}\sum_{cyc}2a^2bc}=\sqrt{3abc(a+b+c)}\geq\sqrt3abc.$$