prove that $a^2+b^2+c^2 \geq a+b+c$

Question

Let $a,b,c$ are positive real numbers such that $abc=1,$

prove that $$a^2+b^2+c^2 \geq a+b+c$$

i was thinking of using AM GM inequlaity.but donot have idea on which pairs to apply it.

any hint....

Answer 1

By QM-AM for the first inequality and AM-GM for the second,

$$\frac{a^2+b^2+c^2}{3}\geq\left( \frac{a+b+c}{3} \right)^2\geq\frac{a+b+c}{3}\left( \sqrt[3]{abc} \right)$$

so since $abc=1, a^2+b^2+c^2\geq a+b+c$