Q:

$a^2 + b^2 + c^2 = 3$, prove that $a + b + c \ge ab + bc + ac$

Let $a$, $b$ and $c$ be positive numbers such that $a^2 + b^2 + c^2 = 3$. Prove that: $$a + b + c \ge ab + bc + ac.$$

Using AM-GM inequality I proved, that $ab + bc + ac \le 3$
I have no idea about moving forward.
Any help is appreciated!

A:

You have $$(a+b+c)^2 \geq 3(ab+bc+ca) = (a^2+b^2+c^2)(ab+bc+ca) \geq (ab+bc+ca)^2$$
So, $$a+b+c\geq ab+bc+ca.$$

