Question Number 149731 by ajfour last updated on 06/Aug/21 Commented by ajfour last updated on 06/Aug/21 $${If}\:\:{AC}={BC}\:,\:{and} \\ $$$$\:{red}\:{length}={blue}\:{length}=\mathrm{3}, \\ $$$${find}\:{AB}. \\ $$$$ \\ $$…
Question Number 18655 by Tinkutara last updated on 26/Jul/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{integers} \\ $$$$\mathrm{between}\:\mathrm{30},\mathrm{000}\:\mathrm{and}\:\mathrm{80},\mathrm{000}\:\mathrm{in}\:\mathrm{which}\:\mathrm{no} \\ $$$$\mathrm{digit}\:\mathrm{is}\:\mathrm{repeated}. \\ $$ Commented by mrW1 last updated on 27/Jul/17 $$\mathrm{let}'\mathrm{s}\:\mathrm{say}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{XZZZY} \\…
Question Number 84188 by jagoll last updated on 10/Mar/20 $$\mathrm{if}\:\mathrm{2x}+\mathrm{3y}\:=\:\mathrm{2020}? \\ $$$$\mathrm{find}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{3x}+\mathrm{2y}\:\mathrm{for}\:\mathrm{x}\:\mathrm{and}\:\mathrm{natural} \\ $$$$\mathrm{number} \\ $$ Commented by jagoll last updated on 10/Mar/20 $$\mathrm{this}\:\mathrm{diopthantine}\:\mathrm{equation}\:? \\…
Question Number 18653 by Tinkutara last updated on 26/Jul/17 $$\mathrm{Solve}\:\mathrm{the}\:\mathrm{inequality},\:\mid{x}\:−\:\mathrm{1}\mid\:+\:\mid{x}\:+\:\mathrm{1}\mid\:<\:\mathrm{4} \\ $$ Answered by ajfour last updated on 26/Jul/17 $$\mathrm{If}\:\mathrm{x}<−\mathrm{1} \\ $$$$−\left(\mathrm{x}−\mathrm{1}\right)−\left(\mathrm{x}+\mathrm{1}\right)<\mathrm{4} \\ $$$$\Rightarrow\:\mathrm{2x}>−\mathrm{4}\:\:\mathrm{or}\:\:\mathrm{x}>\:−\mathrm{2} \\…
Question Number 18652 by Tinkutara last updated on 26/Jul/17 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{natural}\:\mathrm{number}\:{n}, \\ $$$$\mathrm{the}\:\mathrm{fraction}\:\frac{\mathrm{21}{n}\:+\:\mathrm{4}}{\mathrm{14}{n}\:+\:\mathrm{3}}\:\mathrm{is}\:\mathrm{in}\:\mathrm{its}\:\mathrm{lowest}\:\mathrm{term}. \\ $$ Answered by diofanto last updated on 27/Jul/17 $$\mathrm{Lets}\:\mathrm{prove}\:\mathrm{by}\:\mathrm{contradiction}. \\ $$$$\mathrm{Suppose}\:\mathrm{there}\:\mathrm{is}\:{d}\:>\:\mathrm{1}\:\mathrm{such}\:\mathrm{that} \\…
Question Number 149720 by cherokeesay last updated on 06/Aug/21 Answered by mr W last updated on 06/Aug/21 $$\left({R}−\mathrm{1}\right)^{\mathrm{2}} +\left({R}−\mathrm{2}\right)^{\mathrm{2}} ={R}^{\mathrm{2}} \\ $$$${R}^{\mathrm{2}} −\mathrm{6}{R}+\mathrm{5}=\mathrm{0} \\ $$$$\left({R}−\mathrm{5}\right)\left({R}−\mathrm{1}\right)=\mathrm{0}…
Question Number 18650 by thukada last updated on 26/Jul/17 $$\int{dx}/{x}\sqrt{{x}^{\mathrm{4}} −\mathrm{1}} \\ $$ Answered by sma3l2996 last updated on 26/Jul/17 $${A}=\int\frac{{dx}}{{x}\sqrt{{x}^{\mathrm{4}} −\mathrm{1}}} \\ $$$${let}\:\:{t}=\sqrt{{x}^{\mathrm{4}} −\mathrm{1}}\Rightarrow{dt}=\frac{\mathrm{2}{x}^{\mathrm{3}}…
Question Number 84182 by mr W last updated on 11/Mar/20 $${Find}\:{the}\:{number}\:{of}\:{solutions}\:{for} \\ $$$${positive}\:{integers}\:\left({x},{y},{z}\right)\:{satisfying} \\ $$$$\boldsymbol{{x}}+\mathrm{2}\boldsymbol{{y}}+\mathrm{3}\boldsymbol{{z}}=\boldsymbol{{n}}. \\ $$ Answered by mr W last updated on 10/Mar/20…
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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…