Question Number 142085 by Rexzie last updated on 30/May/21 $${Proof}\:{that}\:\mathrm{1}+\mathrm{3}{n}<{n}^{\mathrm{2}} \:{for}\:{every}\:{positive}\:{integer}\:{n}\geqslant\mathrm{4} \\ $$ Answered by MJS_new last updated on 26/May/21 $$\mathrm{it}'\mathrm{s}\:\mathrm{wrong}\:\mathrm{for}\:\frac{\mathrm{3}−\sqrt{\mathrm{13}}}{\mathrm{2}}\leqslant{n}\leqslant\frac{\mathrm{3}+\sqrt{\mathrm{13}}}{\mathrm{2}} \\ $$ Answered by…
Question Number 11009 by Joel576 last updated on 06/Mar/17 $$\mathrm{If}\:\:{x}^{\mathrm{3}} \:=\:{y}^{\mathrm{3}} \\ $$$$\mathrm{Is}\:\mathrm{it}\:\mathrm{always}\:\mathrm{true}\:\mathrm{that}\:{x}\:=\:{y}\:? \\ $$ Answered by mrW1 last updated on 06/Mar/17 $${yes}.\:{function}\:{f}\left({x}\right)={x}^{\mathrm{3}} \:{is}\:{unique}. \\…
Question Number 142061 by iloveisrael last updated on 26/May/21 $$\sqrt{\mathrm{5}{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}+\mathrm{2}{xy}−\mathrm{4}{xz}+\mathrm{10}}\:+ \\ $$$$\mid\mathrm{2}{x}−{y}−\mathrm{13}\mid\:=\:\mathrm{3}\: \\ $$ Commented by MJS_new last updated on 26/May/21 $$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{question}?…
Question Number 76531 by benjo 1/2 santuyy last updated on 28/Dec/19 $${what}\:{is}\:{a}\:{and}\:{b}\:{such}\:{that}\:{a}+{b}\:=\:{a}×{b}\: \\ $$$${and}\:{a}+{b}\:={a}/{b}\:? \\ $$ Commented by MJS last updated on 28/Dec/19 $$\left(\mathrm{1}\right)\:{a}+{b}={ab}\:\Rightarrow\:{a}=\frac{{b}}{{b}−\mathrm{1}}\:\Rightarrow\:{b}\neq\mathrm{1} \\…
Question Number 142045 by Engr_Jidda last updated on 25/May/21 Answered by Dwaipayan Shikari last updated on 25/May/21 $${y}=\left(\frac{\mathrm{1}}{{x}}\right)^{{x}} \\ $$$${log}\left({y}\right)=−{xlog}\left({x}\right)\Rightarrow\frac{{y}'}{{y}}=−\mathrm{1}−{log}\left({x}\right)\Rightarrow{y}=−{x}^{−{x}} \left(\mathrm{1}+{log}\left({x}\right)\right) \\ $$ Commented by…
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Question Number 10970 by Joel576 last updated on 04/Mar/17 $${x}^{{x}^{{x}^{\iddots^{\mathrm{2}} } } } \:=\:\mathrm{2} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}\:? \\ $$ Answered by ridwan balatif last updated on…
Question Number 142025 by mnjuly1970 last updated on 25/May/21 Answered by som(math1967) last updated on 25/May/21 $$\left(\boldsymbol{{a}}+\boldsymbol{{b}}+\boldsymbol{{c}}\right)^{\mathrm{2}} =\mathrm{1} \\ $$$$\mathrm{2}\left(\boldsymbol{{ab}}+\boldsymbol{{bc}}+\boldsymbol{{ca}}\right)=\mathrm{1}−\mathrm{2} \\ $$$$\boldsymbol{{ab}}+\boldsymbol{{bc}}+\boldsymbol{{ca}}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\boldsymbol{{a}}^{\mathrm{3}} +\boldsymbol{{b}}^{\mathrm{3}}…
Question Number 10944 by 314159 last updated on 03/Mar/17 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{ordered}\:\mathrm{pairs}\:\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{so}\:\mathrm{that}\:\frac{\mathrm{ab}}{\mathrm{a}+\mathrm{b}}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer}. \\ $$$$\left(\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{integers}\right). \\ $$ Commented by FilupS last updated on 03/Mar/17 $${n}=\frac{{ab}}{{a}+{b}},\:\:\:\:\:{n},\:{a},\:{b}\:\in\:\mathbb{Z} \\ $$$$\: \\…
Question Number 10933 by Joel576 last updated on 02/Mar/17 $$\mathrm{If}\:\:{p}\:\:\mathrm{and}\:\:{q}\:\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{for}\:\mathrm{the} \\ $$$${x}^{\mathrm{2}} \:−\:\left({a}\:+\:\mathrm{1}\right){x}\:+\:\left(−{a}\:−\:\frac{\mathrm{5}}{\mathrm{2}}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{The}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\:{p}^{\mathrm{2}\:} +\:{q}^{\mathrm{2}} \:\:\mathrm{is}\:… \\ $$ Answered by ridwan balatif last updated…