Question Number 21236 by Joel577 last updated on 17/Sep/17 $$\mathrm{If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{a}}\:+\:\frac{\mathrm{1}}{\mathrm{3}{a}}\:=\:\frac{\mathrm{1}}{{b}^{\mathrm{2}} \:−\:\mathrm{2}{b}} \\ $$$${a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{Find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{a}\:+\:{b} \\ $$ Answered by mrW1 last updated on 17/Sep/17 $$\frac{\mathrm{1}}{\mathrm{a}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}\right)=\frac{\mathrm{1}}{\mathrm{b}\left(\mathrm{b}−\mathrm{2}\right)}…
Question Number 21234 by Tinkutara last updated on 17/Sep/17 $$\mathrm{Let}\:{f}\left({x}\right)\:=\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c},\:\mathrm{where}\:{a},\:{b},\:{c} \\ $$$$\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{2}{a}, \\ $$$${a}\:+\:{b},\:\mathrm{and}\:{c}\:\mathrm{are}\:\mathrm{all}\:\mathrm{integers},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{integral}\:\mathrm{values}\:\mathrm{between}\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{5}\:\mathrm{that}\:{f}\left({x}\right)\:\mathrm{can}\:\mathrm{take}\:\mathrm{is} \\ $$ Answered by Tinkutara last…
Question Number 21230 by Tinkutara last updated on 16/Sep/17 $$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:{n},\:\mathrm{consider} \\ $$$$\mathrm{the}\:\mathrm{highest}\:\mathrm{common}\:\mathrm{factor}\:{h}_{{n}} \:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:{n}!\:+\:\mathrm{1}\:\mathrm{and}\:\left({n}\:+\:\mathrm{1}\right)!.\:\mathrm{For}\:{n}\:<\:\mathrm{100}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{h}_{{n}} . \\ $$ Answered by dioph last updated…
Question Number 21219 by Tinkutara last updated on 16/Sep/17 $$\left(\mathrm{A}\right)\:\mathrm{If}\:\mid{w}\mid\:=\:\mathrm{2},\:\mathrm{then}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$${z}\:=\:{w}\:−\:\frac{\mathrm{1}}{{w}}\:\mathrm{is}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{If}\:\mid{w}\mid\:=\:\mathrm{1},\:\mathrm{then}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$${z}\:=\:{w}\:+\:\frac{\mathrm{1}}{{w}}\:\mathrm{is}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{Options}\:\mathrm{for}\:\mathrm{both}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}: \\ $$$$\left(\mathrm{p}\right)\:\mathrm{An}\:\mathrm{ellipse}\:\mathrm{with}\:\mathrm{eccentricity}\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\left(\mathrm{q}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mathrm{Im}\:{z} \\ $$$$=\:\mathrm{0} \\…
Question Number 152281 by john_santu last updated on 27/Aug/21 $$\mathrm{Find}\:\mathrm{a}\:\mathrm{triple}\:\mathrm{of}\:\mathrm{rational}\: \\ $$$$\mathrm{numbers}\:\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\sqrt[{\mathrm{3}}]{\sqrt[{\mathrm{3}}]{\mathrm{2}}−\mathrm{1}}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{a}}\:+\sqrt[{\mathrm{3}}]{\mathrm{b}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{c}}\: \\ $$ Answered by puissant last updated on 27/Aug/21 $${t}=\sqrt[{\mathrm{3}}]{\mathrm{2}}\:\Rightarrow\:{t}^{\mathrm{3}} =\mathrm{2}\:\rightarrow\:{t}^{\mathrm{3}}…
Question Number 86741 by M±th+et£s last updated on 30/Mar/20 $$\begin{cases}{{x}+\mathrm{10}{y}+\mathrm{50}{z}=\mathrm{500}}\\{{x}+{y}+{z}=\mathrm{100}}\end{cases} \\ $$$$ \\ $$$${find}\:{x},{y},{z} \\ $$ Commented by mr W last updated on 30/Mar/20 $${x},{y},{z}\in{N}\:?…
Question Number 86737 by M±th+et£s last updated on 30/Mar/20 $${prove}\:{that} \\ $$$$\mathrm{1}/{cos}\mathrm{2}{x}+{cosx}+\mathrm{1}=\frac{{sin}\frac{\mathrm{5}{x}}{\mathrm{2}}}{\mathrm{2}{sin}\frac{{x}}{\mathrm{2}}}+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{2}/\frac{{cos}\left({x}\right)+{isin}\left({x}\right)−\mathrm{1}}{{cos}\left({x}\right)+{isin}\left({x}\right)+\mathrm{1}}=−{i}\:{tan}\left({x}\right) \\ $$$$ \\ $$$$\mathrm{3}/\frac{{cos}\left(\mathrm{5}{x}\right)+{isin}\left(\mathrm{5}{x}\right)+\mathrm{1}}{{cos}\left(\mathrm{5}{x}\right)−{isin}\left({x}\right)+\mathrm{1}}={cos}\left(\mathrm{5}{x}\right)+{isin}\left(\mathrm{5}{x}\right) \\ $$ Commented by som(math1967) last updated…
Question Number 21200 by Tinkutara last updated on 15/Sep/17 $$\mathrm{Suppose}\:\mathrm{an}\:\mathrm{integer}\:{x},\:\mathrm{a}\:\mathrm{natural} \\ $$$$\mathrm{number}\:{n}\:\mathrm{and}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:{p} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{44}{x}\:+\:\mathrm{12}\:=\:{p}^{{n}} . \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{p}. \\ $$ Commented by mrW1 last updated…
Question Number 152265 by mathdanisur last updated on 26/Aug/21 $$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\mathrm{sin}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{k}}\:-\:\sqrt{\mathrm{k}\:-\:\mathrm{1}}}{\:\sqrt{\mathrm{k}\left(\mathrm{k}\:+\:\mathrm{1}\right.}}\right)\:=\:? \\ $$ Answered by mindispower last updated on 27/Aug/21 $${sin}^{−} \left({a}\right)−{sin}^{−} \left({b}\right)={sin}^{−}…
Question Number 152266 by mathdanisur last updated on 26/Aug/21 Answered by qaz last updated on 27/Aug/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\mathrm{dx} \\ $$$$=\left[\left(\mathrm{1}+\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)−\left(\mathrm{1}+\mathrm{x}\right)\right]\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\mid_{\mathrm{0}} ^{\mathrm{1}} +\int_{\mathrm{0}}…