<s>[INST] True or false: $a^2+b^2+c^2 +2abc+1\geq 2(ab+bc+ca)$ Is this inequality true?

$a^2+b^2+c^2 +2abc+1\ge2(ab+bc+ca)$, where $a,b,c\gt0$.

Can you find a counterexample for this or not? 
 [/INST] Let $a=x^3$, $b=y^3$ and $c=z^3$.
Hence, by AM-GM and Schur we obtain:
$$a^2+b^2+c^2+2abc+1\geq a^2+b^2+c^2+3\sqrt{a^2b^2c^2}=\sum_{cyc}(x^6+x^2y^2z^2)\geq$$
$$\geq\sum_{cyc}(x^4y^2+y^4x^2)\geq2\sum_{cyc}x^3y^3=2(ab+ac+bc)$$
and we are done!
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