<s>[INST] Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$. 
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that
  $a^2 + b^2 + c^2 \ge a + b + c$.

I'm supposed to prove this use AM-GM, but can't figure it out. Any hints?
 [/INST] 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 $abc=1, a^2+b^2+c^2\geq a+b+c$
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