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

$$\mathrm{In}△\mathrm{ABC}\mathrm{the}\mathrm{following}\mathrm{relationship}\mathrm{holds}:$$$$\sum _{\mathbf{cyc}}\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2+{\mathrm{c}}^2}+4\prod _{\mathbf{cyc}}\cos \mathrm{A}⩽2$$

Answered by behi834171 last updated on 17/Sep/22

$$\frac{a^2}{b^2+c^2}=\frac{b^2+c^2-2bc.cosA}{b^2+c^2}=$$$$=1-\frac{2bc}{b^2+c^2}.cosA⩽1$$$$\mathrm{\Pi }cosA⩽{\left(\frac{1}{2}\right)}^3=\frac{1}{8}$$$$\Rightarrow lhs=3-\mathrm{\Sigma }\frac{2bc}{b^2+c^2}.cosA+4\mathrm{\Pi }cosA⩽$$$$⩽3-3\times 1+4\times \frac{1}{8}=\frac{1}{2}<<2$$

Commented by Shrinava last updated on 18/Sep/22

$$\mathrm{cool}\mathrm{dera}\mathrm{professor}\mathrm{thank}\mathrm{you}$$

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