<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] $a^2+b^2+c^2-ab-bc-ca\\ 
=(a, b, c)\begin{pmatrix}1&-1/2&-1/2\\-1/2& 1&-1/2\\-1/2&-1/2&1\end{pmatrix}\left(\begin{array}{c}a\\b\\c\end{array}\right)$
$=(a, b, c)A\begin{pmatrix}a\\b\\c\end{pmatrix}$
It is sufficient to prove $A$ is a positive semi-definite.
It follows from that the minor determinant of $A$ is one of $1,\dfrac{3}{4},0$
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