<s>[INST] Prove that $\sqrt{a^2+(1-b)^2}+\sqrt{b^2+(1-c)^2}+\sqrt{c^2+(1-a)^2}\geq\frac{3\sqrt{2}}{2}$ 
Prove that the following inequality
  $$\sqrt{a^2+(1-b)^2}+\sqrt{b^2+(1-c)^2}+\sqrt{c^2+(1-a)^2}\geq\frac{3\sqrt{2}}{2}$$
  holds for arbitrary real numbers $a$, $b$ and $c.$

Someone says, "It's very easy problem. It can also be proved by AM–GM inequality." But I can't reason out the answer to this question.
 [/INST] Hint:
$$\sqrt{\frac{a^2+(1-b)^2}{2}}\ge \frac{|a| + |1-b|}{2}$$
Add up, RHS $\ge \frac{|a|+|1-a|+|b|+|1-b|+|c|+|1-c|}{2}\ge \frac{3}{2}$, on LHS it's our sum divided by $\sqrt{2}$.
</s>