In-ABC-the-following-relationship-holds-cyc-a-2-b-2-c-2-4-cyc-cos-A-2-

Question Number 176365 by Shrinava last updated on 16/Sep/22

In △ ABC the following relationship holds :

\sum_{cyc} \frac{a^{2}}{b^{2} + c^{2}} + 4 \prod_{cyc} \cos A ⩽ 2

Answered by behi834171 last updated on 17/Sep/22

\frac{a^{2}}{b^{2} + c^{2}} = \frac{b^{2} + c^{2} - 2 b c . c o s A}{b^{2} + c^{2}} =

= 1 - \frac{2 b c}{b^{2} + c^{2}} . c o s A ⩽ 1

Π c o s A ⩽ {(\frac{1}{2})}^{3} = \frac{1}{8}

\Rightarrow l h s = 3 - Σ \frac{2 b c}{b^{2} + c^{2}} . c o s A + 4 Π c o s A ⩽

⩽ 3 - 3 \times 1 + 4 \times \frac{1}{8} = \frac{1}{2} << 2

Commented by Shrinava last updated on 18/Sep/22

cool dera professor thank you