Question Number 104516 by Quvonchbek last updated on 22/Jul/20 Commented by mr W last updated on 22/Jul/20 $${Red}\:{length}={r}_{\mathrm{1}} \pi+{r}_{\mathrm{2}} \pi=\left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)\pi \\ $$$$={R}\pi={blue}\:{length} \\…
Question Number 170053 by mathlove last updated on 15/May/22 Answered by greougoury555 last updated on 15/May/22 $$\:\:\frac{\mathrm{6}{x}}{{x}−\mathrm{2}}\:−\sqrt{\frac{\mathrm{12}{x}}{{x}−\mathrm{2}}}−\mathrm{2}\:\sqrt[{\mathrm{4}}]{\frac{\mathrm{12}{x}}{{x}−\mathrm{2}}}\:\geqslant\:\mathrm{0} \\ $$$$\:{let}\:\sqrt[{\mathrm{4}}]{\frac{\mathrm{12}{x}}{{x}−\mathrm{2}}}\:=\:{u}\:;\:{u}\geqslant\mathrm{0} \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{\mathrm{2}}{u}^{\mathrm{4}} −{u}^{\mathrm{2}} −\mathrm{2}{u}\:\geqslant\:\mathrm{0} \\ $$$$\Rightarrow{u}^{\mathrm{4}}…
Question Number 104517 by Quvonchbek last updated on 22/Jul/20 Answered by 1549442205PVT last updated on 22/Jul/20 $$\mathrm{It}\:\mathrm{is}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{see}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{numbers}: \\ $$$$\mathrm{11},\mathrm{19},\mathrm{29},\mathrm{41},…\mathrm{has}\:\mathrm{general}\:\mathrm{term}\:\mathrm{is} \\ $$$$\mathrm{n}^{\mathrm{2}} +\mathrm{3n}+\mathrm{1}\left(\mathrm{n}\in\mathbb{N}^{\ast} \right).\mathrm{Hence}, \\ $$$$\mathrm{S}=\underset{\mathrm{n}=\mathrm{1}}…
Question Number 38933 by MrW3 last updated on 01/Jul/18 $$\left[{Refreshing}\:{an}\:{old}\:{question}\right] \\ $$$${Find}\:{the}\:{smallest}\:{number}\:{which}\:{is} \\ $$$${a}\:{multiple}\:{of}\:\mathrm{75}\:{and}\:{has}\:\mathrm{75}\:{divisors}. \\ $$ Commented by math1967 last updated on 01/Jul/18 $$\mathrm{75}=\mathrm{3}×\mathrm{5}^{\mathrm{2}} \:\therefore{factor}\:{of}\:\mathrm{75}…
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Question Number 169989 by udaythool last updated on 13/May/22 $$ \\ $$$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}: \\ $$$$\mathrm{For}\:\mathrm{any}\:\mathrm{extension}\:{E}\:\mathrm{of}\:\mathrm{a}\:\mathrm{field}\:{F}, \\ $$$${F}\left({u}\right)={F}\left[{u}\right]\:\:\:\:\forall\:{u}\in{E}. \\ $$$$\mathrm{Where}\:{F}\left({u}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{smallest}\:\mathrm{subfield} \\ $$$$\mathrm{of}\:{E}\:\mathrm{containing}\:{F}\:\mathrm{and}\:{u}\:\mathrm{and} \\ $$$${F}\left[{u}\right]=\left\{{f}\left({u}\right)\mid{f}\left({x}\right)\in{F}\left[{x}\right]\right\},\:{F}\left[{x}\right]\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{polynomial}\:\mathrm{ring}\:\mathrm{over}\:{F}. \\…
Question Number 104450 by Anindita last updated on 21/Jul/20 $$\frac{\mathrm{4}}{\mathrm{99}}\:+\:\frac{\mathrm{7}}{\mathrm{999}}\:+\:\frac{\mathrm{11}}{\mathrm{999999}}\:=\:? \\ $$$$\boldsymbol{\mathrm{Can}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{this}}? \\ $$ Answered by OlafThorendsen last updated on 21/Jul/20 $$\mathrm{999999}\:=\:\mathrm{999000}+\mathrm{999}\:=\:\mathrm{999}×\mathrm{1001} \\ $$$$\mathrm{999999}\:=\:\mathrm{990000}+\mathrm{9900}+\mathrm{99}\:=\:\mathrm{99}×\mathrm{10101} \\…
Question Number 38907 by ajfour last updated on 01/Jul/18 $$\sqrt{{c}^{\mathrm{2}} +{x}^{\mathrm{2}} }\:=\:\mathrm{1}+\frac{{c}}{{x}} \\ $$$${how}\:{many}\:{complex}\:{roots}\:? \\ $$ Commented by ajfour last updated on 01/Jul/18 Commented by…
Question Number 169977 by infinityaction last updated on 13/May/22 Answered by ajfour last updated on 13/May/22 $$\left({a}+{b}+{c}+{d}\right)^{\mathrm{2}} \geqslant\mathrm{16}\sqrt{{abcd}} \\ $$$$\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \right)\geqslant\mathrm{4}\sqrt{{abcd}} \\…
Question Number 104419 by Anindita last updated on 21/Jul/20 $$\boldsymbol{\mathrm{Now}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{father}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{is}} \\ $$$$\mathrm{90}\:\boldsymbol{\mathrm{years}}.\:\mathrm{5}\:\boldsymbol{\mathrm{years}}\:\boldsymbol{\mathrm{ago}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{father}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{age}} \\ $$$$\boldsymbol{\mathrm{was}}\:\mathrm{3}\:\boldsymbol{\mathrm{times}}\:\boldsymbol{\mathrm{than}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{age}}.\:\boldsymbol{\mathrm{Write}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{age}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{father}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{age}}\:\mathrm{3}\:\boldsymbol{\mathrm{years}} \\ $$$$\boldsymbol{\mathrm{after}}. \\ $$ Answered by 1549442205PVT last updated…