Inequality(open ended)[made by shanzeed anwar]

[nafistiham]

posting on request.

Let $a_1,a_2,a_3,.....,a_n$ are positive real numbers.
prove or disprove it,
if,
\[a_1+a_2+a_3+.....+a_n=n\]
then,
\[\sum_{cyclic}^{} \frac{a_1}{1+a_2^2}\geq \frac {n}{2}\]

Example:
$\sum_{cyclic}^{} \frac{a}{1+b^2}$ for $a,b,c$ is equal to
\[\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\]

[SANZEED]

It's open ended because the equality is only my conjecture.

[Phlembac Adib Hasan]

I think it may be not true for all $n$s.Re-arrangement inequality gives me such a result that may not be true for large $n$s.It's very ugly so I'm not posting it here.Now I 'll try to find out such a case that does not follow this inequality.