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Question Number 144742 by loveineq last updated on 28/Jun/21

Let a,b,c>0 and a+b+c = 3. Prove that                        ((((ab)/(ab+1))+((bc)/(bc+1))+((ca)/(ca+1)))/((1/(ab+1))+(1/(bc+1))+(1/(ca+1)))) ≥ abc  (Found by WolframAlpha and                  inspired by my old problem)

$$\mathrm{Let}\:{a},{b},{c}>\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that}\:\: \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\frac{{ab}}{{ab}+\mathrm{1}}+\frac{{bc}}{{bc}+\mathrm{1}}+\frac{{ca}}{{ca}+\mathrm{1}}}{\frac{\mathrm{1}}{{ab}+\mathrm{1}}+\frac{\mathrm{1}}{{bc}+\mathrm{1}}+\frac{\mathrm{1}}{{ca}+\mathrm{1}}}\:\geqslant\:{abc} \\ $$ $$\left(\mathrm{Found}\:\mathrm{by}\:\mathrm{WolframAlpha}\:\mathrm{and}\:\right. \\ $$ $$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{inspired}\:\mathrm{by}\:\mathrm{my}\:\mathrm{old}\:\mathrm{problem}\right) \\ $$

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