<s>[INST] How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$? How can the following inequation be proven?
$$a^2 + b^2 + c^2 \ge ab + bc + ca$$
 [/INST] If $\ c> a ,a^2+c^2 \gt 2ac $, because $\ (a-c)^2 \gt 0$
If $\ c>b>a ,c^2+b^2/2+a^2/2 \gt ac+bc $ and $\ b^2/2+a^2/2 \gt ab $, sum of them $\ a^2+b^2+c^2 \gt ab+bc+ac $
If $\ c=b \gt a $ or $\ a=b=c $, it can be solved with same logic.
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