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Question Number 33681 by naka3546 last updated on 22/Apr/18

Let  a, b, c   are  positive real  numbers  such  that   (1/a) + (1/b) + (1/c)  =  3 .  Prove that :   a + b + c  +  (4/(1 + (((abc)^2 ))^(1/3) ))   ≥  5

$${Let}\:\:{a},\:{b},\:{c}\:\:\:{are}\:\:{positive}\:{real}\:\:{numbers}\:\:{such}\:\:{that}\:\:\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:\:=\:\:\mathrm{3}\:. \\ $$$${Prove}\:{that}\::\:\:\:{a}\:+\:{b}\:+\:{c}\:\:+\:\:\frac{\mathrm{4}}{\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\left({abc}\right)^{\mathrm{2}} }}\:\:\:\geqslant\:\:\mathrm{5} \\ $$

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