Question Number 19053 by Tinkutara last updated on 03/Aug/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}:\:\frac{\mathrm{1}}{\mathrm{2}}\:−\:\frac{\mathrm{1}}{\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\:−\:\frac{\mathrm{1}}{\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$−\:\frac{\mathrm{1}}{\mathrm{7}}\:+\:\frac{\mathrm{1}}{\mathrm{8}}\:−\:\frac{\mathrm{1}}{\mathrm{9}}\:+\:… \\ $$ Answered by ajfour last updated on 03/Aug/17 $$\mathrm{ln}\:\left(\mathrm{1}+\mathrm{x}\right)=\mathrm{x}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{3}}−\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{4}}+\frac{\mathrm{x}^{\mathrm{5}}…
Question Number 19009 by chux last updated on 03/Aug/17 $$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{four}\:\mathrm{consecutive} \\ $$$$\mathrm{2}−\mathrm{digit}\:\mathrm{numbers}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\mathrm{10}\:\mathrm{becomes}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}.\mathrm{Which} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{can}\:\mathrm{possibly}\:\mathrm{be} \\ $$$$\mathrm{one}\:\mathrm{of}\:\mathrm{these}\:\mathrm{four}\:\mathrm{numbers}? \\ $$$$\left(\mathrm{a}\right)\mathrm{21}\left(\mathrm{b}\right)\mathrm{25}\left(\mathrm{c}\right)\mathrm{41}\left(\mathrm{d}\right)\mathrm{67}\left(\mathrm{e}\right)\mathrm{73} \\ $$$$ \\ $$$$ \\…
Question Number 84532 by john santu last updated on 14/Mar/20 $$\mathrm{x}>\:\mathrm{0}\:,\:\mathrm{y}\:>\:\mathrm{0}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{xy}}{\mathrm{x}+\mathrm{y}}\:<\:\mathrm{x} \\ $$ Commented by mr W last updated on 14/Mar/20 $$\Rightarrow{y}<{x}+{y} \\…
Question Number 84461 by Tony Lin last updated on 13/Mar/20 $${p}^{\mathrm{2}} +\mathrm{3}{q}^{\mathrm{2}} =\mathrm{11907},\:{p},{q}\in\mathbb{Z},{find}\:{p\&q} \\ $$ Commented by mr W last updated on 13/Mar/20 $${i}\:{got}\:{only}\:{two}\:{solutions}: \\…
Question Number 84288 by jagoll last updated on 11/Mar/20 Commented by jagoll last updated on 11/Mar/20 $$\mathrm{dear}\:\mathrm{Mr}\:\mathrm{W}.\:\mathrm{please}\:\mathrm{help}\:\mathrm{me} \\ $$ Commented by jagoll last updated on…
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…