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Question Number 82514 by naka3546 last updated on 22/Feb/20 $${Find}\:\:\:{maximum}\:\:{of}\:\:{n}\:\:{such}\:\:{that}\:\: \\ $$$${n}^{\mathrm{2}} \:+\:{n}\:+\:\mathrm{1}\:\:{is}\:\:{factor}\:\:{of}\:\:\mathrm{19}{n}^{\mathrm{2019}} \:+\:\mathrm{1}\:. \\ $$ Commented by naka3546 last updated on 22/Feb/20 $${n}\:\:\in\:\:\mathbb{N} \\…

Question Number 148051 by abdurehime last updated on 25/Jul/21 $$\rightharpoondown\left(\rightharpoondown\mathrm{p}\rightarrow\mathrm{q}\right)\vee\left(\mathrm{p}\wedge\rightharpoondown\mathrm{q}\right)\: \\ $$$$\mathrm{simplify} \\ $$ Commented by abdurehime last updated on 25/Jul/21 $$\mathrm{please}\:\mathrm{sir}\:\mathrm{help}\:\mathrm{me} \\ $$ Commented…

Question Number 82485 by naka3546 last updated on 21/Feb/20 Commented by abdomathmax last updated on 21/Feb/20 $${let}\:{f}\left({x}\right)=\frac{\mathrm{6}}{{t}^{\mathrm{2}} }−\frac{{sin}\left(\mathrm{6}{t}\right)}{{t}^{\mathrm{3}} {cos}^{\mathrm{2}} \left(\mathrm{3}{t}\right)}\:\Rightarrow \\ $$$${f}\left({x}\right)=\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\left\{\mathrm{6}−\frac{{sin}\left(\mathrm{6}{t}\right)}{{t}\:{cos}^{\mathrm{2}} \left(\mathrm{3}{t}\right)}\right\}\:{we}\:{have}\: \\…

Question Number 82456 by liki last updated on 21/Feb/20 Commented by abdomathmax last updated on 21/Feb/20 $${cos}\left(\mathrm{5}{x}\right)={sin}\left(\mathrm{4}{x}\right)\Leftrightarrow{cos}\left(\mathrm{5}{x}\right)={cos}\left(\frac{\pi}{\mathrm{2}}−\mathrm{4}{x}\right)\:\Rightarrow \\ $$$$\mathrm{5}{x}=\frac{\pi}{\mathrm{2}}−\mathrm{4}{x}\:+\mathrm{2}{k}\pi\:{or}\:\mathrm{5}{x}=−\frac{\pi}{\mathrm{2}}\:+\mathrm{4}{x}\:+\mathrm{2}{k}\pi\:\Rightarrow \\ $$$$\mathrm{9}{x}\:=\frac{\pi}{\mathrm{2}}\:+\mathrm{2}{k}\pi\:{or}\:{x}=−\frac{\pi}{\mathrm{2}}\:+\mathrm{2}{k}\pi\:\:\left({k}\in{Z}\right)\:\Rightarrow \\ $$$${x}=\frac{\pi}{\mathrm{18}}\:+\frac{\mathrm{2}{k}\pi}{\mathrm{9}}\:{or}\:{x}=−\frac{\pi}{\mathrm{2}}\:+\mathrm{2}{k}\pi \\ $$$${S}\:=\left\{\frac{\pi}{\mathrm{18}}\:+\frac{\mathrm{2}{k}\pi}{\mathrm{9}}\:,{k}\epsilon{Z}\right\}\cup\left\{\left(\mathrm{2}{k}−\frac{\mathrm{1}}{\mathrm{2}}\right)\pi\:,{k}\in{Z}\right\}…

Question Number 82421 by M±th+et£s last updated on 21/Feb/20 Answered by mind is power last updated on 21/Feb/20 $${let}\:{y}\in\left[{a},{b}\right]\:{such}\:{sup}\:{f}={f}\left({y}\right) \\ $$$${let}\:{z}\in\left[{a},{b}\right]\:{such}\:{sup}\:{g}={g}\left({z}\right) \\ $$$${if}\:{z}={y}\Rightarrow{c}={y}\Rightarrow{f}\left({c}\right)={g}\left({c}\right) \\ $$$${if}\:{z}#{y}\:{assum}\:{z}<{y}…