Question Number 195369 by Shlock last updated on 31/Jul/23 Answered by witcher3 last updated on 01/Aug/23 $$\mathrm{x}=−\mathrm{y} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{f}\left(\mathrm{0}\right)\right)=\mathrm{2f}\left(\mathrm{x}^{\mathrm{2}} \right) \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{2}}\mathrm{f}\left(\mathrm{f}\left(\mathrm{0}\right)\right)=\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right) \\ $$$$\forall\left(\mathrm{x},\mathrm{y}\right)\in\mathbb{Z}\:^{\mathrm{2}}…
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Question Number 195331 by Shlock last updated on 30/Jul/23 Answered by mr W last updated on 06/Aug/23 $${say}\:{a}={pc},\:{b}={qc} \\ $$$$\left({p}+{q}+\mathrm{1}\right)\left(\frac{\mathrm{1}}{{p}}+\frac{\mathrm{1}}{{q}}+\mathrm{1}\right)=\mathrm{10} \\ $$$$\frac{{a}+{b}}{{c}}={p}+{q} \\ $$$$\left({p}+{q}\right)_{{max}} =?…
Question Number 195330 by Rupesh123 last updated on 30/Jul/23 Commented by AST last updated on 30/Jul/23 $$\Rightarrow\left({abc}\right)^{\mathrm{3}} =\mathrm{10}^{\mathrm{45}} \Rightarrow{abc}=\mathrm{10}^{\mathrm{15}} \\ $$ Terms of Service Privacy…
Question Number 195292 by Rupesh123 last updated on 29/Jul/23 Answered by MM42 last updated on 29/Jul/23 $$\frac{\left({y}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{4}}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}=\mathrm{1}\:\:\Rightarrow{O}'\left(\mathrm{0},\mathrm{1}\right)\:,\:\:{A}\left(\mathrm{0},\mathrm{3}\right)\:\:,\:\:{A}'\left(\mathrm{0},−\mathrm{1}\right) \\ $$$$\Rightarrow{min}\left\{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right\}=\mathrm{1}\:\checkmark\: \\ $$…
Question Number 195291 by Matica last updated on 29/Jul/23 $$\:\:{it}\:{is}\:{given}\:{a},{b},{c}\:\in\:\mathbb{N}^{\ast} \:\:{and}\:\:{ab}<{c}\:.\:{Prove}\:{that}\:{a}+{b}\leqslant{c}. \\ $$ Answered by Frix last updated on 29/Jul/23 $${c}={ab}+\mathrm{1} \\ $$$${a}+{b}\leqslant{ab}+\mathrm{1} \\ $$$$\mathrm{Let}\:{a}<{b}…
Question Number 195288 by CrispyXYZ last updated on 29/Jul/23 $${x},\:{y},\:{z}\in\mathbb{R}_{+} , \\ $$$${P}\:=\:\frac{{x}}{{x}\:+\:{y}}\:+\:\frac{{y}}{{y}\:+\:{z}}\:+\:\frac{{z}}{{z}\:+\:{x}}, \\ $$$${Q}\:=\:\frac{{y}}{{x}\:+\:{y}}\:+\:\frac{{z}}{{y}\:+\:{z}}\:+\:\frac{{x}}{{z}\:+\:{x}}, \\ $$$${Q}\:=\:\frac{{z}}{{x}\:+\:{y}}\:+\:\frac{{x}}{{y}\:+\:{z}}\:+\:\frac{{y}}{{z}\:+\:{x}}. \\ $$$${f}\left({x},\:{y},\:{z}\right)=\mathrm{max}\left\{{P},\:{Q},\:{R}\right\},\:\mathrm{find}\:{f}_{\mathrm{min}} . \\ $$ Commented by Frix…
Question Number 195229 by mathlove last updated on 28/Jul/23 $${below}\:{equestion}\:{is}\:{show}\:\:{elips}\:{and} \\ $$$${hypharabollah} \\ $$$$\frac{{x}^{\mathrm{2}} }{{cos}\mathrm{3}}+\frac{{y}^{\mathrm{2}} }{{sin}\mathrm{3}}=\mathrm{1} \\ $$ Commented by mr W last updated on…