<s>[INST] show this inequality $ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$ 
let $a,b,c>0$ and such $a+b+c=3abc$, show that
  $$ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$$
                         Proposed by wang yong xi

since
$$\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}=3$$
so use Cauchy-Schwarz inequality we have
$$(ab+bc+ac)\left(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}\right)\ge 9$$
so we have
$$ab+bc+ac\ge 3$$
 [/INST] Hint : Prove with Jensen's inequality (apply to the function $f(x)=0.5x^2+\sin(x-1)$ ) this with the obvious restriction $ab+bc+ca\leq6$:

$$ab+bc+ca+\sin(a-1)+\sin(b-1)+\sin(c-1)$$$$\geq$$$$ ab+bc+ca-0.5(a^2+b^2+c^2)+3\sin(\frac{a+b+c}{3}-1)+1.5(\frac{a+b+c}{3})^2$$

And now a last hint prove this (with the condition $3abc=a+b+c$ and $ab+bc+ca\leq6$ ):

$$ab+bc+ca-0.5(a^2+b^2+c^2)+1.5(\frac{a+b+c}{3})^2\geq 3$$
  And 
  $$3\sin(\frac{a+b+c}{3}-1)\geq 0$$

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