Question Number 18663 by Tinkutara last updated on 26/Jul/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{101}\:×\:\mathrm{10001}\:× \\ $$$$\mathrm{100000001}\:×\:…\:×\:\left(\mathrm{1000}…\mathrm{01}\right)\:\mathrm{where}\:\mathrm{the} \\ $$$$\mathrm{last}\:\mathrm{factor}\:\mathrm{has}\:\mathrm{2}^{\mathrm{7}} \:−\:\mathrm{1}\:\mathrm{zeros}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{ones}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ones}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{product}. \\ $$ Commented by diofanto last…
Question Number 149718 by peter frank last updated on 06/Aug/21 Answered by Ar Brandon last updated on 06/Aug/21 $${I}=\int\mathrm{cos2}\theta\mathrm{ln}\left(\frac{\mathrm{cos}\theta+\mathrm{sin}\theta}{\mathrm{cos}\theta−\mathrm{sin}\theta}\right){d}\theta \\ $$$$\:\:=\int\mathrm{cos2}\theta\mathrm{ln}\left(\frac{\mathrm{1}+\mathrm{tan}\theta}{\mathrm{1}−\mathrm{tan}\theta}\right){d}\theta=\int\mathrm{cos2}\theta\mathrm{ln}\left(\mathrm{tan}\left(\frac{\pi}{\mathrm{4}}+\theta\right)\right){d}\theta \\ $$$$\begin{cases}{{u}\left({x}\right)=\mathrm{ln}\left(\mathrm{tan}\left(\frac{\pi}{\mathrm{4}}+\theta\right)\right)}\\{{v}'\left({x}\right)=\mathrm{cos2}\theta}\end{cases}\Rightarrow\begin{cases}{{u}'\left({x}\right)=\frac{\mathrm{2}}{\mathrm{sin}\left(\frac{\pi}{\mathrm{2}}+\mathrm{2}\theta\right)}}\\{{v}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin2}\theta}\end{cases} \\ $$$${I}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin2}\theta\centerdot\mathrm{ln}\left(\mathrm{tan}\left(\frac{\pi}{\mathrm{4}}+\theta\right)\right)−\int\frac{\mathrm{sin2}\theta}{\mathrm{cos2}\theta}{d}\theta…
Question Number 18385 by Tinkutara last updated on 19/Jul/17 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{integers}\:\mathrm{which}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{11}\:\mathrm{times}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{digits}. \\ $$ Answered by mrW1 last updated on 19/Jul/17 $$\mathrm{let}'\mathrm{s}\:\mathrm{see}\:\mathrm{if}\:\mathrm{such}\:\mathrm{a}\:\mathrm{number}\:\mathrm{can}\:\mathrm{have}\:\mathrm{only} \\ $$$$\mathrm{one}\:\mathrm{digit}: \\…
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Question Number 18214 by chux last updated on 16/Jul/17 $$\mathrm{The}\:\mathrm{speeds}\:\mathrm{of}\:\mathrm{Daniel}\:\mathrm{and}\:\mathrm{Robert} \\ $$$$\mathrm{are}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{3}:\mathrm{4}.\mathrm{In}\:\mathrm{a}\:\mathrm{race}\:\mathrm{of} \\ $$$$\mathrm{300m}\:\mathrm{Daniel}\:\mathrm{has}\:\mathrm{a}\:\mathrm{start}\:\mathrm{of}\:\mathrm{90m}.\: \\ $$$$\mathrm{Daniel}\:\mathrm{won}\:\mathrm{by}? \\ $$ Answered by ajfour last updated on 17/Jul/17…
Question Number 18213 by chux last updated on 16/Jul/17 $$\mathrm{The}\:\mathrm{greatest}\:\mathrm{number}\:\mathrm{that}\:\mathrm{will} \\ $$$$\mathrm{divide}\:\mathrm{82}\:,\mathrm{111}\:\mathrm{and}\:\mathrm{140}\:\mathrm{leaving}\:\mathrm{the}\: \\ $$$$\mathrm{same}\:\mathrm{remainder}\:\mathrm{in}\:\mathrm{each}\:\mathrm{case}\:\mathrm{is}…….. \\ $$ Answered by Tinkutara last updated on 17/Jul/17 $$\mathrm{82}\:=\:{xa}\:+\:{r} \\…
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Question Number 83629 by M±th+et£s last updated on 04/Mar/20 $${show}\:{that} \\ $$$$\underset{{n},{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}!\:{k}!}{\left({n}+{k}+\mathrm{2}\right)!}=\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$ Answered by Kamel Kamel last updated on 04/Mar/20…
Question Number 149061 by jlewis last updated on 02/Aug/21 $$\mathrm{factorise}\:\:\:\:\mathrm{4}/\mathrm{5}^{\mathrm{x}} +\mathrm{1} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 17767 by Joel577 last updated on 10/Jul/17 $$\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{30}} {\sum}}\left({n}^{\mathrm{2}} \:+\:\mathrm{1}\right)\:=\: \\ $$$$\left(\mathrm{A}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{224}\right) \\ $$$$\left(\mathrm{B}\right)\:\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{15}} {\sum}}\left(\mathrm{2}{n}^{\mathrm{2}} \:+\:\mathrm{30}{n}\:+\:\mathrm{225}\right) \\ $$$$\left(\mathrm{C}\right)\:\underset{{n}\:=\:\mathrm{1}}…