Question Number 221017 by Rojarani last updated on 22/May/25 Answered by Ghisom last updated on 23/May/25 $$\mathrm{min}\:\left(\frac{{a}^{\mathrm{2}} }{{b}}+\frac{{b}^{\mathrm{2}} }{{c}}+\frac{{c}^{\mathrm{2}} }{{a}}\right)\:=\mathrm{1}\:\mathrm{when}\:{a}={b}={c}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\mathrm{min}\:\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)\:=\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{when}\:{a}={b}={c}=\frac{\mathrm{1}}{\mathrm{3}}…
Question Number 220987 by fantastic last updated on 21/May/25 $${Let}\:{a},{b},{c}\:{be}\:{positive}\:{reals}\:{such}\:{that}\:{abc}=\mathrm{1}.{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} \left({b}+{c}\right)}+\frac{\mathrm{1}}{{b}^{\mathrm{3}} \left({c}+{a}\right)}+\frac{\mathrm{1}}{{c}^{\mathrm{3}} \left({a}+{b}\right)}\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$ Answered by mr W last updated on 21/May/25…
Question Number 220958 by mr W last updated on 21/May/25 $${for}\:{x},\:{y},\:{z}\:>\mathrm{0}\:{find}\:{the}\:{maximum}\:{of} \\ $$$${x}^{{m}} {y}^{{n}} {z}^{{k}} \:{subject}\:{to}\:{ax}+{by}+{cz}={d}. \\ $$ Commented by mr W last updated on…
Question Number 220947 by fantastic last updated on 21/May/25 $$\underset{{k}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}\:\:\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+\frac{\left({k}−\mathrm{1}\right)\pi}{\mathrm{6}}\right)\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+\frac{{k}\pi}{\mathrm{6}}\right)} \\ $$ Answered by MrGaster last updated on 21/May/25 $$=\underset{{k}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}\mathrm{2}\left[\mathrm{cot}\left(\frac{\pi}{\mathrm{4}}+\frac{\left({k}−\mathrm{1}\right)\pi}{\mathrm{6}}\right)−\mathrm{cot}\left(\frac{\pi}{\mathrm{4}}+\frac{{k}\pi}{\mathrm{6}}\right)\right] \\…
Question Number 220995 by hardmath last updated on 21/May/25 Answered by Frix last updated on 22/May/25 $$\mathrm{Let}\:\mathrm{the}\:\mathrm{red}\:\mathrm{line}\:=\mathrm{1} \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{30}°}=\frac{{a}}{\mathrm{sin}\:\mathrm{15}°}\:\Rightarrow \\ $$$$\:\:\:\:\:{a}=\frac{\sqrt{\mathrm{6}}−\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\frac{{a}}{\mathrm{sin}\:{x}}=\frac{\mathrm{1}}{\mathrm{sin}\:\left(\mathrm{135}°−{x}\right)}=\frac{\sqrt{\mathrm{2}}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}\Rightarrow \\ $$$$\:\:\:\:\:{a}=\frac{\sqrt{\mathrm{2}}\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}=\frac{\sqrt{\mathrm{2}}\mathrm{tan}\:{x}}{\mathrm{1}+\mathrm{tan}\:{x}}…
Question Number 220858 by Rojarani last updated on 20/May/25 Commented by mr W last updated on 21/May/25 $$\mathrm{65}? \\ $$ Commented by Rojarani last updated…
Question Number 220853 by fantastic last updated on 20/May/25 $${The}\:{two}\:{solutions}\:{of}\:{the}\:{equation}\:{are}\:{the}\:{same} \\ $$$${a}\left({b}−{c}\right){x}^{\mathrm{2}\:} +{b}\left({c}−{a}\right){x}+{c}\left({a}−{b}\right)=\mathrm{0} \\ $$$${Prove}\:{that}\:\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{c}}=\frac{\mathrm{2}}{{b}} \\ $$ Answered by fantastic last updated on 20/May/25 $${In}\:{any}\:{quadratic}\:{equation}\:\alpha{x}^{\mathrm{2}}…
Question Number 220854 by fantastic last updated on 20/May/25 $${Solve}\:{for}\:{x}\:\:\:{and}\:\:\:\:{y} \\ $$$$\mathrm{3}^{{x}} +\mathrm{3}^{{y}} =\mathrm{4},\:\:\mathrm{3}^{−{x}} +\mathrm{3}^{−{y}\:} =\frac{\mathrm{4}}{\mathrm{3}} \\ $$ Answered by SdC355 last updated on 20/May/25…
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Question Number 220810 by Rojarani last updated on 19/May/25 Commented by Ghisom last updated on 19/May/25 $$\mathrm{without}\:\mathrm{further}\:\mathrm{information}\:\mathrm{we}\:\mathrm{can}\:\mathrm{let} \\ $$$${a}={b}={c}={k}\:\Rightarrow \\ $$$${k}=\frac{\mathrm{2}^{\mathrm{1}/\mathrm{3}} −\mathrm{1}}{\mathrm{27}}\:\Rightarrow \\ $$$${a}+{b}+{c}=\mathrm{3}{k}=\frac{\mathrm{2}^{\mathrm{1}/\mathrm{3}} −\mathrm{1}}{\mathrm{9}}…