Question Number 21247 by Tinkutara last updated on 17/Sep/17 $$\mathrm{If}\:\left[\:\right]\:\mathrm{represents}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer} \\ $$$$\mathrm{function}\:\mathrm{and}\:{f}\left({x}\right)\:=\:{x}\:−\:\left[{x}\right]\:\mathrm{then} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${f}\left({x}\right)\:+\:{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{1}\:\mathrm{are}\:\mathrm{infinite}. \\ $$$$\boldsymbol{\mathrm{True}}/\boldsymbol{\mathrm{False}} \\ $$ Answered by dioph last updated…
Question Number 152314 by mathdanisur last updated on 27/Aug/21 Answered by ghimisi last updated on 27/Aug/21 $$\frac{{a}^{\mathrm{2}} }{{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} }+\frac{{c}^{\mathrm{2}} }{{c}\left({a}+{b}+{c}\right)}\geqslant\frac{\left({a}+{c}\right)^{\mathrm{2}} }{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{ab}+{bc}+{ac}}…
Question Number 21235 by Tinkutara last updated on 17/Sep/17 $$\mathrm{For}\:\mathrm{any}\:\mathrm{integer}\:{k},\:\mathrm{let}\:\alpha_{{k}} \:=\:\mathrm{cos}\:\left(\frac{{k}\pi}{\mathrm{7}}\right)\:+ \\ $$$${i}\:\mathrm{sin}\:\left(\frac{{k}\pi}{\mathrm{7}}\right),\:\mathrm{where}\:{i}\:=\:\sqrt{−\mathrm{1}}.\:\mathrm{The}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\frac{\underset{{k}=\mathrm{1}} {\overset{\mathrm{12}} {\sum}}\mid\alpha_{{k}+\mathrm{1}} \:−\:\alpha_{{k}} \mid}{\underset{{k}=\mathrm{1}} {\overset{\mathrm{3}} {\sum}}\mid\alpha_{\mathrm{4}{k}−\mathrm{1}} \:−\:\alpha_{\mathrm{4}{k}−\mathrm{2}} \mid}\:\mathrm{is} \\ $$…
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…