Question Number 16528 by I'm a gamer last updated on 23/Jun/17
Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 24/Jun/17
$$\mathrm{3}\left({a}+{b}+{c}\right)−\frac{{a}}{{b}}−\frac{{b}}{{c}}−\frac{{c}}{{a}}\leqslant\mathrm{6} \\ $$$$\mathrm{3}\geqslant\mathrm{3}\sqrt[{\mathrm{3}}]{{abc}\sqrt{{abc}}}=\mathrm{3}\left({abc}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \Rightarrow{abc}\leqslant\mathrm{1} \\ $$$${a}+{b}+{c}\leqslant\mathrm{3}\sqrt[{\mathrm{3}}]{{abc}}\leqslant\mathrm{3} \\ $$$$\Rightarrow{LHS}\leqslant\mathrm{3}\left(\mathrm{3}\right)−\mathrm{3}=\mathrm{6}\leqslant\mathrm{6} \\ $$