https://artofproblemsolving.com/community/c4h2626456_cota_cotb_cotca2b2c24s__1993_kyiv_city_mo_105 here i already proved that $$\cot A+ \cot B + \cot C = \frac{a^2+b^2+c^2}{4S}.$$ so enouth to show $$\cot A+ \cot B + \cot C \ge \sqrt{3}$$if $\triangle{ABC}$ is acute then $\cot(\theta)$ is a convex function ..........so be Jensen's inequality $$\frac{(\cot A+ \cot B + \cot C)}{3} \ge \cot(\frac{A+B+C}{3}) = \cot(\frac{\pi}{3}) = \frac{1}{\sqrt{3}}$$which completes the proof....

I have another proof using heron, but I don't know latex,also I don't know how to attach files Sorry

@above, here I have instructions on how to add an attachment in aops

Here is the solution Thanks @above

Attachments:

The last inequality is easy to prove with am gm