<s>[INST] Prove for all real numbers $a, \ b,$ and $c$, $bc + ac + ab \leq a^2 + b^2 + c^2$. I've been having some trouble with this particular problem. Could someone provide a hint of some sort? My original attempt that turned out to be useless is to start off with
$$(b+c)^2 + (a+c)^2 + (a+b)^2 = (b+c)^2 + (a+c)^2 + (a+b)^2$$
and subtract from both sides until I get the above inequality, but I found out I'd end up subtracting that very same inequality and would be right back where I started.
 [/INST] $
a^2 + b^2 + c^2 \ge ab + bc + ca\\
\Longleftrightarrow \frac12(2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca) \ge 0\\
\Longleftrightarrow (a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (c^2 - 2ac + a^2) \ge 0 \\
\Longleftrightarrow (a-b)^2 +(b-c)^2 +(c-a)^2 \ge 0
$
which is obviously true. Giving a hint for this problem is as good as solving it entirely. :-) 
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