Question Number 125413 by Hassen_Timol last updated on 11/Dec/20 $$\mathrm{Knowing}\:\mathrm{that}\:: \\ $$$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}^{\mathrm{2}} \:=\:\begin{pmatrix}{\mathrm{2}{n}}\\{\:\:{n}}\end{pmatrix} \\ $$$$ \\ $$$$\mathrm{We}\:\mathrm{have}\:{p}\:\mathrm{tokens}\:\mathrm{and}\:{n}\:\mathrm{boxes}. \\ $$$$\mathrm{Each}\:\mathrm{box}\:\mathrm{is}\:\mathrm{labeled}\:\mathrm{with}\:\mathrm{a}\:\mathrm{number}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:{n}. \\ $$$$\mathrm{Each}\:\mathrm{box}\:\mathrm{is}\:\mathrm{enough}\:\mathrm{big}\:\mathrm{to}\:\mathrm{receive}\:\mathrm{all}\:{p}\:\mathrm{tokens}. \\ $$$$…
Question Number 59861 by ANTARES VY last updated on 15/May/19 $$\sqrt{\boldsymbol{{a}}+\boldsymbol{{b}}\sqrt{\boldsymbol{{c}}}}=\sqrt{\frac{\boldsymbol{{a}}+\sqrt{\boldsymbol{{a}}^{\mathrm{2}} −\boldsymbol{{b}}^{\mathrm{2}} \boldsymbol{{c}}}}{\mathrm{2}}}+\sqrt{\frac{\boldsymbol{{a}}−\sqrt{\boldsymbol{{a}}^{\mathrm{2}} −\boldsymbol{{b}}^{\mathrm{2}} \boldsymbol{{c}}}}{\mathrm{2}}}. \\ $$$$\boldsymbol{{prove}} \\ $$ Commented by Kunal12588 last updated on…
Question Number 59858 by jimful last updated on 15/May/19 $$\mathrm{P}\left(\mathrm{1}\right)=\frac{\mathrm{1}}{{i}}\: \\ $$$$\mathrm{P}_{\mathrm{n}+\mathrm{1}} \mathrm{P}_{\mathrm{n}} =\mathrm{1}−\mathrm{P}_{\mathrm{n}+\mathrm{1}} \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}Im}\left(\mathrm{P}_{\mathrm{n}} \right)\:=? \\ $$ Terms of Service Privacy Policy…
Question Number 190928 by TUN last updated on 14/Apr/23 $${Solve}\:{the}\:{equation}\:: \\ $$$$\mathrm{2}+{x}^{\mathrm{2}} \left({x}^{\mathrm{4}} +\mathrm{1}\right)=\frac{\sqrt{\mathrm{2}}{x}^{\mathrm{3}} \left({x}^{\mathrm{4}} −\mathrm{1}\right)}{\:\sqrt{{x}^{\mathrm{4}} +\mathrm{1}}} \\ $$$$ \\ $$ Commented by Frix last…
Question Number 190925 by sonukgindia last updated on 14/Apr/23 Commented by Frix last updated on 14/Apr/23 $${x}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$ Answered by ibrahimabdullayev last updated on…
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Question Number 59829 by ANTARES VY last updated on 15/May/19 Commented by ANTARES VY last updated on 15/May/19 $$\boldsymbol{\mathrm{Find}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{painted}}\:\:\boldsymbol{\mathrm{area}} \\ $$ Commented by MJS last…
Question Number 125359 by weltr last updated on 10/Dec/20 $$\begin{cases}{{x}\:+\:{y}\:=\:\frac{\mathrm{5}\pi}{\mathrm{3}}}\\{{sin}\:{x}\:=\:\mathrm{2}{sin}\:{y}}\end{cases} \\ $$ Answered by Ar Brandon last updated on 10/Dec/20 $$\begin{cases}{\mathrm{x}+\mathrm{y}=\frac{\mathrm{5}\pi}{\mathrm{3}}}&{…\left(\mathrm{i}\right)}\\{\mathrm{sinx}=\mathrm{2siny}}&{…\left(\mathrm{ii}\right)}\end{cases} \\ $$$$\mathrm{From}\:\left(\mathrm{i}\right),\:\mathrm{x}+\mathrm{y}=\frac{\mathrm{5}\pi}{\mathrm{3}}\:\Rightarrow\:\mathrm{y}=\frac{\mathrm{5}\pi}{\mathrm{3}}−\mathrm{x} \\ $$$$\mathrm{substituting}\:\mathrm{the}\:\mathrm{above}\:\mathrm{result}\:\mathrm{in}\:\left(\mathrm{ii}\right)\:\mathrm{we}\:\mathrm{have}\:;…
Question Number 59814 by ANTARES VY last updated on 15/May/19 Commented by ANTARES VY last updated on 15/May/19 $$\boldsymbol{\mathrm{F}}\mathrm{ind}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{painted}}\:\:\boldsymbol{\mathrm{area}} \\ $$ Answered by mr W…
Question Number 59812 by ANTARES VY last updated on 15/May/19 $$\frac{\mathrm{6}}{\mathrm{3}+\sqrt[{\mathrm{3}}]{\mathrm{3}}+\sqrt[{\mathrm{3}}]{\mathrm{9}}}\:\:\:\boldsymbol{\mathrm{simplify}}. \\ $$ Commented by Kunal12588 last updated on 15/May/19 $${let}\:\sqrt[{\mathrm{3}}]{\mathrm{3}}={a} \\ $$$$\Rightarrow\sqrt[{\mathrm{3}}]{\mathrm{9}}={a}^{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{3}={a}^{\mathrm{3}}…