Question Number 198132 by a.lgnaoui last updated on 11/Oct/23 $${Solve}: \\ $$$$\frac{\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{7}\boldsymbol{\mathrm{x}}−\mathrm{5}\right)}{\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}+\mathrm{2}\right)}=\mathrm{2} \\ $$ Answered by som(math1967) last updated on 11/Oct/23 $${log}\left({x}^{\mathrm{2}} +\mathrm{7}{x}−\mathrm{5}\right)=\mathrm{2}{log}\left({x}+\mathrm{2}\right) \\…
Question Number 198131 by a.lgnaoui last updated on 11/Oct/23 $$\mathrm{Resoudre} \\ $$$$\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}−\mathrm{3}\right)+\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}−\mathrm{2}\right)=\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{x}}−\mathrm{21}\right) \\ $$$$ \\ $$ Answered by Rasheed.Sindhi last updated on 11/Oct/23 $$\mathrm{log}\left(\mathrm{x}−\mathrm{3}\right)+\mathrm{log}\left(\mathrm{x}−\mathrm{2}\right)=\mathrm{log}\left(\mathrm{x}^{\mathrm{2}}…
Question Number 198152 by universe last updated on 11/Oct/23 $$\:\:\:\:{a}_{{n}+\mathrm{2}} \:=\:\:\:\sqrt{{a}_{{n}} ×{a}_{{n}+\mathrm{1}} }\:\:\:\forall\:{n}\geqslant\mathrm{1}\:,\:{n}\:\in\:\mathrm{N} \\ $$$$\:\mathrm{and}\:\mathrm{here}\:\:\mathrm{a}_{\mathrm{1}\:} =\:\alpha\:\:{and}\:{a}_{\mathrm{2}} =\:\beta\:\:\mathrm{then} \\ $$$$\:\:\:\mathrm{prove}\:\mathrm{that}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}+\mathrm{2}} \:\:\:=\:\:\left(\alpha×\beta^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{3}} \\ $$ Answered…
Question Number 198151 by sonukgindia last updated on 11/Oct/23 Answered by Mathspace last updated on 12/Oct/23 $${I}=\int_{\mathrm{0}} ^{\infty} \:\frac{{zlnz}}{\mathrm{1}+{z}^{\mathrm{3}} }{dz}\:\:\:\:\left({z}={t}^{\frac{\mathrm{1}}{\mathrm{3}}} \right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{\frac{\mathrm{1}}{\mathrm{3}}}…
Question Number 198145 by Erico last updated on 11/Oct/23 $$\mathrm{Soit}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2x}−\mathrm{1}}{\mathrm{lnx}−\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}\:\mathrm{et}\:\mathrm{I}=\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$$$\mathrm{1}.\:\mathrm{Montrer}\:\mathrm{que}\:\forall\mathrm{n}\geqslant\mathrm{2} \\ $$$$\:\:\:\frac{\mathrm{1}}{\mathrm{4n}}\:\leqslant\:\mathrm{I}\:−\:\frac{\pi}{\mathrm{2n}^{\mathrm{3}} }\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\:\frac{\mathrm{k}\left(\mathrm{n}−\mathrm{k}\right)}{\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{n}}\right)}\:\leqslant\:\frac{\mathrm{3}}{\mathrm{4n}} \\ $$$$\mathrm{2}.\:\mathrm{En}\:\mathrm{d}\acute {\mathrm{e}duire}\:\mathrm{la}\:\mathrm{valeur}\:\mathrm{de}\:\mathrm{I} \\ $$$$ \\…
Question Number 198146 by necx122 last updated on 11/Oct/23 $${Please}\:{suggest}\:{youtube}\:{playlist}\:{to} \\ $$$${prepare}\:{one}\:{for}\:{mathematics}\:{olympiad}. \\ $$$${Thanks}\:{in}\:{advance}. \\ $$$$ \\ $$ Commented by TheHoneyCat last updated on 13/Oct/23…
Question Number 198147 by mr W last updated on 11/Oct/23 $${if}\:{a},{x},{y},{b}\:{is}\:{an}\:{AP}\:{and}\:{a},{p},{q},{b}\:{is}\:{a}\:{GP}. \\ $$$${prove}\:{that}\:{xy}\geqslant{pq}. \\ $$$$\left({with}\:{a},\:{b}\:>\mathrm{0}\right) \\ $$ Answered by AST last updated on 11/Oct/23 $${p}^{\mathrm{2}}…
Question Number 198077 by SANOGO last updated on 10/Oct/23 Answered by Mathspace last updated on 10/Oct/23 $${par}\:{recurrence}\:{sur}\:{n} \\ $$$${on}\:\Phi\left(\mathrm{0}\right)\geqslant\mathrm{0}\:{vraie}\:{puisque}\:\Phi\left({N}\right)\subset{N} \\ $$$${supposons}\:{que}\:\Phi\left({n}\right)\geqslant{n}\:{et}\:{montrons} \\ $$$${que}\:\Phi\left({n}+\mathrm{1}\right)\geqslant{n}+\mathrm{1} \\ $$$${on}\:{n}+\mathrm{1}>{n}\:{et}\:\Phi\:{strictement}\:…
Question Number 198104 by sonu753 last updated on 10/Oct/23 Answered by mr W last updated on 11/Oct/23 $$\frac{{dx}}{{dy}}−\frac{{x}}{{y}}=−{y} \\ $$$$\left[{d}.{e}.\:{of}\:{type}\:{y}'+{p}\left({x}\right){y}={q}\left({x}\right)\right] \\ $$$$\int{p}\left({y}\right){dy}=−\int\frac{{dy}}{{y}}=−\mathrm{ln}\:{y} \\ $$$${u}\left({y}\right)={e}^{−\mathrm{ln}\:{y}} =\frac{\mathrm{1}}{{y}}…
Question Number 198103 by mr W last updated on 10/Oct/23 $${solve}\:{for}\:{x},\:{y}\:\in{N} \\ $$$$\sqrt{{x}}+\sqrt{{y}}=\sqrt{\mathrm{2023}} \\ $$ Answered by Safojon last updated on 10/Oct/23 $$\sqrt{\mathrm{7}}+\mathrm{16}\sqrt{\mathrm{7}}=\sqrt{\mathrm{2023}} \\ $$$$\mathrm{2}\sqrt{\mathrm{7}}+\mathrm{15}\sqrt{\mathrm{7}}=\sqrt{\mathrm{2023}}…