Question Number 141454 by loveineq last updated on 19/May/21
$$\mathrm{Let}\:{a},{b},{c}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{4}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} }{\left({a}+{b}+\mathrm{4}\right)^{\mathrm{2}} }+\frac{{b}^{\mathrm{2}} +{bc}+{c}^{\mathrm{2}} }{\left({b}+{c}+\mathrm{4}\right)^{\mathrm{2}} }+\frac{{c}^{\mathrm{2}} +{ca}+{a}^{\mathrm{2}} }{\left({c}+{a}+\mathrm{4}\right)^{\mathrm{2}} }\:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$