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Question Number 132292 by Algoritm last updated on 13/Feb/21 Answered by mathmax by abdo last updated on 13/Feb/21 $$\frac{\mathrm{1}}{\left[\mathrm{x}\right]}+\frac{\mathrm{1}}{\left[\mathrm{2x}\right]}=\left[\mathrm{x}\right]+\frac{\mathrm{1}}{\mathrm{3}}\:\:\mathrm{let}\:\left[\mathrm{x}\right]=\mathrm{n}\:\Rightarrow\mathrm{n}\leqslant\mathrm{x}<\mathrm{n}+\mathrm{1}\:\Rightarrow\mathrm{2n}\leqslant\mathrm{2x}<\mathrm{2n}+\mathrm{2} \\ $$$$\mathrm{if}\:\mathrm{2x}\in\left[\mathrm{2n},\mathrm{2n}+\mathrm{1}\left[\:\Rightarrow\mathrm{x}\in\left[\mathrm{n},\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\left[\:\Rightarrow\left[\mathrm{2x}\right]=\mathrm{2n}\right.\right.\right.\right. \\ $$$$\mathrm{e}\Rightarrow\frac{\mathrm{1}}{\mathrm{n}}+\frac{\mathrm{1}}{\mathrm{2n}}=\mathrm{n}+\frac{\mathrm{1}}{\mathrm{3}}\:\Rightarrow\frac{\mathrm{3}}{\mathrm{2n}}=\mathrm{n}+\frac{\mathrm{1}}{\mathrm{3}}\:\Rightarrow\frac{\mathrm{9}}{\mathrm{2n}}=\mathrm{3n}+\mathrm{1}\:\Rightarrow\mathrm{9}=\mathrm{2n}\left(\mathrm{3n}+\mathrm{1}\right)\:\Rightarrow \\ $$$$\mathrm{6n}^{\mathrm{2}}…

Question Number 66759 by aliesam last updated on 19/Aug/19 Commented by mr W last updated on 19/Aug/19 $${question}\:{is}\:{wrong}. \\ $$$${there}\:{is}\:{no}\:{x}\:{which}\:{fulfills}\:\mid\frac{\mathrm{sin}\:{x}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\mid>\mathrm{2}, \\ $$$${but}\:{such}\:{that}\:{e}^{{x}} >\mathrm{2016},\:{x}>\mathrm{ln}\:\mathrm{2016}\approx\mathrm{7}.\mathrm{6} \\…

Question Number 66751 by John Kaloki Musau last updated on 19/Aug/19 $${A}\:{rostum}\:{is}\:{made}\:{by}\:{cutting}\:{off} \\ $$$${the}\:{upper}\:{part}\:{of}\:{a}\:{cone}\:{along}\:{a} \\ $$$${plane}\:{parallel}\:{to}\:{the}\:{base}\:{at}\:\frac{\mathrm{2}}{\mathrm{3}}\:{up}\: \\ $$$${the}\:{height}.\:{What}\:{fraction}\:{of}\:{the}\: \\ $$$${volume}\:{of}\:{the}\:{cone}\:{does}\:{the} \\ $$$${rostum}\:{represent}? \\ $$ Answered…

Question Number 132261 by muneer0o0 last updated on 12/Feb/21 Answered by Olaf last updated on 13/Feb/21 $$\Omega\:=\:\int_{−\mathrm{1}} ^{+\mathrm{1}} \int_{−\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }} ^{+\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }} \mathrm{sec}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy}…

Question Number 66718 by otchereabdullai@gmail.com last updated on 18/Aug/19 Commented by Tony Lin last updated on 19/Aug/19 Commented by Tony Lin last updated on 19/Aug/19…