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Let-a-b-c-gt-0-and-a-b-c-3-Prove-that-ab-1-ab-c-bc-1-bc-a-ca-1-ca-b-a-b-c-




Question Number 144118 by loveineq last updated on 21/Jun/21
Let a,b,c > 0 and a+b+c = 3. Prove that                  (((√(ab))+1)/( (√(ab))+(√c)))+(((√(bc))+1)/( (√(bc))+(√a)))+(((√(ca))+1)/( (√(ca))+(√b))) ≥ (√a)+(√b)+(√c)
$$\mathrm{Let}\:{a},{b},{c}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt{{ab}}+\mathrm{1}}{\:\sqrt{{ab}}+\sqrt{{c}}}+\frac{\sqrt{{bc}}+\mathrm{1}}{\:\sqrt{{bc}}+\sqrt{{a}}}+\frac{\sqrt{{ca}}+\mathrm{1}}{\:\sqrt{{ca}}+\sqrt{{b}}}\:\geqslant\:\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

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