Question Number 127882 by zakirullah last updated on 02/Jan/21 Commented by mr W last updated on 03/Jan/21 $${please}\:{pay}\:{attention}\:{when}\:{typing}! \\ $$$$\left(\mathrm{111}\right)\mathrm{8}\:{is}\:{misleading},\:{since}\:{you} \\ $$$${meant}\:\left(\mathrm{111}\right)_{\mathrm{8}} . \\ $$…
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Question Number 127861 by naka3546 last updated on 02/Jan/21 $${How}\:\:{many}\:\:{natural}\:\:{numbers}\:\:{n}\:\leqslant\:\mathrm{2020}\:{such}\:\:{that} \\ $$$$\left({n}\:+\:\mathrm{3}\right)!\:\:{is}\:\:{divisible}\:\:{by}\:\:\mathrm{2}^{{n}} \:? \\ $$ Answered by floor(10²Eta[1]) last updated on 03/Jan/21 $$\mathrm{2}^{\mathrm{n}} \mid\left(\mathrm{n}+\mathrm{3}\right)!\Rightarrow\mathrm{k}\geqslant\mathrm{n} \\…
Question Number 62308 by aliesam last updated on 19/Jun/19 Commented by maxmathsup by imad last updated on 20/Jun/19 $${x}^{\mathrm{2}} −\mathrm{2}{x}\:\:+\mathrm{2}\:=\mathrm{0}\:\rightarrow\Delta^{'} \:=\mathrm{1}−\mathrm{2}\:=−\mathrm{1}\:={i}^{\mathrm{2}} \:\Rightarrow\alpha\:=\mathrm{1}+{i}\:\:{and}\:\beta\:=\mathrm{1}−{i}\:=\overset{−} {\alpha}\:\Rightarrow \\ $$$$\alpha^{{n}}…
Question Number 62289 by naka3546 last updated on 19/Jun/19 Answered by MJS last updated on 19/Jun/19 $$\mathrm{we}\:\mathrm{have} \\ $$$${A}_{\mathrm{1}} =\mathrm{3}=\frac{\mathrm{1}}{\mathrm{2}}×{a}×{x}\:\Rightarrow\:{x}=\frac{\mathrm{6}}{{a}} \\ $$$${A}_{\mathrm{2}} =\mathrm{5}=\frac{\mathrm{1}}{\mathrm{2}}×{b}×{y}\:\Rightarrow\:{y}=\frac{\mathrm{10}}{{b}} \\ $$$$\mathrm{side}\:\mathrm{of}\:\mathrm{triangle}\:=\:{c}…
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Question Number 62273 by alphaprime last updated on 18/Jun/19 $${Mr}.\:{Rasheed}.{Sindhi}\: \\ $$$${I}\:{sense}\:{you}'{re}\:{much}\:{engaged}\:{in}\:{making}\: \\ $$$${olympiad}\:{contents}\:{these}\:{days}\:,\:{I}\:{wish}\: \\ $$$${that}\:{you}\:{join}\:{my}\:{workspace}\:{concerning}\:{that}\:{same}. \\ $$ Commented by Rasheed.Sindhi last updated on 20/Jun/19…
Question Number 62265 by azizullah last updated on 18/Jun/19 Commented by tanmay last updated on 18/Jun/19 $${question}\:{is}\:{not}\:{clear}… \\ $$ Answered by MJS last updated on…
Question Number 127781 by shaker last updated on 02/Jan/21 Answered by bemath last updated on 02/Jan/21 $$\begin{cases}{\mathrm{2sin}\:\mathrm{xcos}\:\mathrm{y}=−\mathrm{1}}\\{\mathrm{2cos}\:\mathrm{xsin}\:\mathrm{y}=\mathrm{1}}\end{cases} \\ $$$$\Rightarrow\mathrm{sin}\:\left(\mathrm{x}+\mathrm{y}\right)+\mathrm{sin}\:\left(\mathrm{x}−\mathrm{y}\right)=−\mathrm{1} \\ $$$$\Rightarrow\mathrm{sin}\:\left(\mathrm{x}+\mathrm{y}\right)−\mathrm{sin}\:\left(\mathrm{x}−\mathrm{y}\right)=\mathrm{1} \\ $$$$\Leftrightarrow\:\mathrm{2sin}\:\left(\mathrm{x}+\mathrm{y}\right)=\mathrm{0}\:\Rightarrow\mathrm{sin}\:\left(\mathrm{x}+\mathrm{y}\right)=\mathrm{0} \\ $$$$\:\Rightarrow\mathrm{x}+\mathrm{y}\:=\:\mathrm{n}\pi\:;\:\mathrm{n}\in\mathbb{Z}\:…
Question Number 127752 by Olaf last updated on 01/Jan/21 $$\mathrm{I}\:\mathrm{wish}\:\mathrm{you}\:\mathrm{an}\:\mathrm{happy}\:\mathrm{new}\:\mathrm{year}\:\mathrm{from}\:\mathrm{Paris}. \\ $$ Commented by mindispower last updated on 01/Jan/21 $${are}\:{you}\:{french}\:{sir}\:? \\ $$ Commented by Dwaipayan…