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Question Number 209341 by Tawa11 last updated on 07/Jul/24 $$\mathrm{solve}\:\:\:\:\mathrm{x}^{\mathrm{log}\:\mathrm{27}} \:\:+\:\:\mathrm{9}^{\mathrm{log}\:\mathrm{x}} \:\:=\:\:\:\mathrm{36} \\ $$ Answered by Frix last updated on 07/Jul/24 $${x}^{\mathrm{log}_{{b}} \:\mathrm{27}} ={x}^{\frac{\mathrm{3ln}\:\mathrm{3}}{\mathrm{ln}\:{b}}} \\…
Question Number 209342 by essaad last updated on 07/Jul/24 Answered by Berbere last updated on 07/Jul/24 $${U}_{{n}+\mathrm{1}} ={f}\left({U}_{{n}} \right) \\ $$$${x}\overset{{f}} {\rightarrow}\frac{{x}}{\mathrm{2}}+\frac{{x}^{\mathrm{2}} }{\mathrm{4}};{f}\:{increase} \\ $$$${f}\left(\left[\mathrm{0},\mathrm{1}\right]\right)=\left[\mathrm{0},\frac{\mathrm{3}}{\mathrm{4}}\right]\subset\left[\mathrm{0},\mathrm{1}\right]…
Question Number 209336 by Tawa11 last updated on 07/Jul/24 Commented by Tawa11 last updated on 07/Jul/24 $$\mathrm{Find}\:\mathrm{area}\:\mathrm{of}\:\mathrm{shadded}. \\ $$ Answered by mr W last updated…
Question Number 209332 by efronzo1 last updated on 07/Jul/24 Answered by Frix last updated on 07/Jul/24 $${x}^{\mathrm{3}} −\mathrm{8}{x}^{\mathrm{2}} +\left(\mathrm{16}−{k}\right){x}=\mathrm{0} \\ $$$${x}_{\mathrm{1}} =\mathrm{0}\:\:{x}_{\mathrm{2}} =\mathrm{4}−\sqrt{{k}}\:\:{x}_{\mathrm{3}} =\mathrm{4}+\sqrt{{k}} \\…
Question Number 209329 by Huy250 last updated on 07/Jul/24 $${Given}\:{an}\:{acute}\:{triangle}\:{with}\:{AB}\:<{AC} \\ $$$${is}\:{inscribed}\:{in}\:{the}\:{circle}\:\left({O}\right).\:{Let}\:{D} \\ $$$${and}\:{E}\:{be}\:{the}\:{midpoints}\:{of}\:{the}\:{minor}\:{arc} \\ $$$${and}\:{major}\:{arc}\:{BC},\:{respectively}.\:{Let}\:{I}\:{and}\:{J} \\ $$$${be}\:{the}\:{incenters}\:{of}\:{trianges}\:{ABD}\:{and}\:{ACD}, \\ $$$${respectively}.\:{Prove}\:{that}\:{EI}={EJ}. \\ $$ Commented by Huy250…
Question Number 209352 by Spillover last updated on 07/Jul/24 Answered by mr W last updated on 08/Jul/24 $${L}={mr}^{\mathrm{2}} \omega={constant} \\ $$$$\omega_{{max}} =\frac{{L}}{{mr}_{{min}} ^{\mathrm{2}} }=\frac{{L}}{{m}\left({a}−{c}\right)^{\mathrm{2}} }…
Question Number 209347 by RoseAli last updated on 07/Jul/24 Answered by Berbere last updated on 07/Jul/24 $$\int_{−\mathrm{4}} ^{\mathrm{4}} {f}\left({x}\right){dx}=\mathrm{2}\underset{\mathrm{0}} {\int}^{\mathrm{4}} {f}\left({x}^{\mathrm{2}} \right){dx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{4}}…
Question Number 209308 by Erico last updated on 06/Jul/24 $$\mathrm{Donner}\:\mathrm{l}'\acute {\mathrm{e}quivalence}\:\mathrm{simple} \\ $$$$\mathrm{de}\:\mathrm{I}_{\mathrm{n}} =\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{{t}^{{n}} }{{t}^{{n}} −{t}+\mathrm{1}}{dt} \\ $$ Answered by mathzup last updated…
Question Number 209309 by hardmath last updated on 06/Jul/24 $$\mathrm{m}\:,\:\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{m}\:\geqslant\:\mathrm{2}\:\:\:\mathrm{and}\:\:\:\mathrm{n}\:\geqslant\:\mathrm{2} \\ $$$$\mathrm{p}\:>\:\mathrm{0}\:\:\:\mathrm{and}\:\:\:\mathrm{q}\:>\:\mathrm{0} \\ $$$$\mathrm{p}\:+\:\mathrm{q}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\left(\mathrm{1}−\mathrm{q}^{\boldsymbol{\mathrm{n}}} \right)^{\boldsymbol{\mathrm{m}}} \:+\:\left(\mathrm{1}−\mathrm{p}^{\boldsymbol{\mathrm{m}}} \right)^{\boldsymbol{\mathrm{n}}} \:\geqslant\:\mathrm{1} \\ $$ Terms…