<s>[INST] If $a+b+c = 6$ and $a$, $b$, $c$ are nonnegative then $a^2+b^2+c^2 \geq 12$ Let $a,b,c$ be three positive real numbers such that $a+b+c = 6$. Prove that $a^2+b^2+c^2 \geq 12$.
I tried using the AM-GM inequality to solve the same, however I wasn't able to make any considerable progress.
 [/INST] Use Cauchy–Schwartz:
$$a+b+c\ \leqslant\ \sqrt{1^2+1^2+1^2}\sqrt{a^2+b^2+c^2}$$
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