Question Number 156867 by cortano last updated on 16/Oct/21 Answered by qaz last updated on 16/Oct/21 $$\frac{\mathrm{1}}{\mathrm{a}+\mathrm{b}}+\frac{\mathrm{9}}{\mathrm{c}}+\frac{\mathrm{16}}{\mathrm{a}+\mathrm{b}+\mathrm{c}}\geqslant\frac{\left(\mathrm{1}+\mathrm{3}+\mathrm{4}\right)^{\mathrm{2}} }{\mathrm{2}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)}=\mathrm{8} \\ $$$$\mathrm{Min}=\mathrm{8}\:\mathrm{holds}\:\mathrm{only}\:\mathrm{when}\:\frac{\mathrm{1}}{\mathrm{a}+\mathrm{b}}=\frac{\mathrm{3}}{\mathrm{c}}=\frac{\mathrm{4}}{\mathrm{a}+\mathrm{b}+\mathrm{c}}\:\:\mathrm{ie}.\mathrm{a}+\mathrm{b}=\mathrm{1}\:\:\:\mathrm{c}=\mathrm{3} \\ $$ Commented by cortano…
Question Number 25787 by ibraheem160 last updated on 14/Dec/17 $$\mathrm{2}^{{x}} ={x}^{\mathrm{2}\:} ,{hence}\:{x}=\mathrm{2}\:.\:{prove} \\ $$ Commented by mrW1 last updated on 14/Dec/17 $${It}'{s}\:{not}\:{true},\:{Sir}. \\ $$$${If}\:\mathrm{2}^{{x}} ={x}^{\mathrm{2}}…
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Question Number 156834 by MathSh last updated on 16/Oct/21 Answered by MJS_new last updated on 16/Oct/21 $${x}=\mathrm{2} \\ $$ Commented by MathSh last updated on…
Question Number 91277 by john santu last updated on 29/Apr/20 $${p}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}+…+\frac{\mathrm{1}}{\mathrm{2003}}−\frac{\mathrm{1}}{\mathrm{2004}} \\ $$$${q}=\frac{\mathrm{1}}{\mathrm{1003}}+\frac{\mathrm{1}}{\mathrm{1004}}+…+\frac{\mathrm{1}}{\mathrm{2004}} \\ $$$${p}^{\mathrm{2}} +{q}^{\mathrm{2}} \:=\: \\ $$ Answered by naka3546 last updated on…
Question Number 156809 by MathSh last updated on 15/Oct/21 $$\mathrm{Solve}\:\mathrm{in}\:\mathbb{R} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4x}\:=\:\sqrt{\mathrm{40x}^{\mathrm{2}} \:+\:\mathrm{32x}\:-\:\mathrm{16}} \\ $$ Commented by MathSh last updated on 15/Oct/21 $$\mathrm{Yes}\:\mathrm{dear}\:\boldsymbol{\mathrm{S}}\mathrm{er} \\…
Question Number 156808 by MathSh last updated on 15/Oct/21 $$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\int\:\frac{\mathrm{x}^{\mathrm{7}} \:-\:\mathrm{x}^{\mathrm{5}} \:+\:\mathrm{x}^{\mathrm{3}} \:-\:\mathrm{x}}{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{10}} }\:\mathrm{dx}\:\:;\:\:\mathrm{x}\in\mathbb{R} \\ $$ Commented by tabata last updated on 15/Oct/21…
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Question Number 156806 by MathSh last updated on 15/Oct/21 Answered by mr W last updated on 16/Oct/21 $$\frac{{AB}}{{AC}}=\frac{\mathrm{sin}\:\mathrm{60}}{\mathrm{sin}\:\mathrm{20}} \\ $$$$\frac{{BC}}{{AC}}=\frac{\mathrm{sin}\:\mathrm{80}}{\mathrm{sin}\:\mathrm{20}} \\ $$$$\frac{{DC}}{{AC}}=\frac{\mathrm{sin}\:\mathrm{80}}{\mathrm{sin}\:\mathrm{30}} \\ $$$$\frac{{AB}×{DC}}{{BC}×{AC}}=\frac{{AB}}{{AC}}×\frac{{AC}}{{BC}}×\frac{{DC}}{{AC}} \\…