<s>[INST] Let $a,b,c$ be non-negative real numbers .Prove that : $ a^2+b^2+c^2 +\sqrt{2} abc + 2\sqrt{2} +3 \geq (2+\sqrt {2} )(a+b+c) $ Let $a,b,c$ be non-negative real numbers
Prove that : $ a^2+b^2+c^2 +\sqrt{2} abc + 2\sqrt{2} +3 \geq (2+\sqrt {2} )(a+b+c) $
My idea is to use the $(p,q,r)$ method:
$p=a+b+c$
$q=ab+bc+ca$
$r = abc $
$\Rightarrow a^2+b^2+c^2 = p^2-2q $
$ a^2+b^2+c^2 +\sqrt{2} abc + 2\sqrt{2} +3 \geq (2+\sqrt {2} )(a+b+c) $
or
$ p^2-2q +\sqrt{2} r + 2\sqrt{2} +3 \geq (2+\sqrt {2} )p $
or
$ (p - (1+\sqrt{2} ))^2 -2q +\sqrt{2} r \geq 0 $
or
The problem I am facing is exactly what I want to prove : $\sqrt{2} r \geq 2q$ is completely wrong .
I hope to get help from everyone. Thanks very much !
 [/INST] Another way.
Since $$\prod_{cyc}(a-1)^2=\prod_{cyc}((b-1)(c-1))\geq0,$$ we can assume $(b-1)(c-1)\geq0$, which gives $$a(b-1)(c-1)\geq0$$ or$$abc\geq ab+ac-a$$ and since $$b^2+c^2\geq\frac{1}{2}(b+c)^2,$$ it's enough to prove that
$$a^2+\frac{1}{2}(b+c)^2+\sqrt2(ab+ac-a)+2\sqrt2+3\geq(2+\sqrt2)(a+b+c)$$ or $$(\sqrt2a+b+c-2-\sqrt2)^2\geq0$$ and we are done!
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