$$(a^2 + b^2 + c^2)/3 \geq ((a + b + c)/3)^2$$ OR $$(a^2 + b^2 + c^2)/3 \leq ((a + b + c)/3)^2$$
for all $a, b, c \geq 0.$ The problem wants me to find which inequality is correct and then provide a proof for it.
I'm pretty sure the first inequality is correct (I assumed this after substituting random values as $a, b,$ and $c$). I've tried factoring the inequalities and this is what I ended up with:
$$9(a^2 + b^2 + c^2) \geq 3(a^2 + b^2 + c^2 - 2ab + 2ac + 2bc)$$
However, I don't know where to go from here. I'm also meant to be applying the AM-GM theorem to solve this problem but I'm unsure where and how to apply it in this situation. Any help would be extremely appreciated :)