If-a-b-c-gt-0-and-a-2-b-2-c-2-abc-Prove-that-a-a-2-bc-b-b-2-ac-c-c-2-ab-1-2-

Question Number 205643 by hardmath last updated on 26/Mar/24

If a, b, c > 0 and a^{2} + b^{2} + c^{2} = abc

Prove that :

\frac{a}{a^{2} + bc} + \frac{b}{b^{2} + ac} + \frac{c}{c^{2} + ab} ⩽ \frac{1}{2}

Answered by A5T last updated on 29/Mar/24

Σ \frac{a}{a^{2} + b c} ⩽ \sqrt{a^{2} + b^{2} + c^{2}} \sqrt{Σ \frac{1}{{(a^{2} + b c)}^{2}}}

⩽ \sqrt{a b c} \times \frac{1}{2 \sqrt{a b c}} (\sqrt{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}) ⩽ \frac{1}{2} (\sqrt{\frac{a b + b c + c a}{a b c}})

= \frac{1}{2} \sqrt{\frac{a b + b c + c a}{a^{2} + b^{2} + c^{2}}} ⩽ \frac{1}{2}

Facebook Tweet Pin