Problem - 3221
If $a>1$, then $$\frac{1}{a-1}+\frac{1}{a} + \frac{1}{a+1} > \frac{3}{a}$$ holds.
The claim is equivalent to $$\frac{1}{a-1}+ \frac{1}{a+1} > \frac{2}{a}$$
By HM-AM inequality: $$\frac{2}{\frac{1}{a-1}+\frac{1}{a+1}}<\frac{(a-1)+(a+1)}{2}=a$$