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Question Number 228019 by mr W last updated on 10/Mar/26 $${t}\:{is}\:{the}\:{fractional}\:{part}\:{of}\:{a},\:{and} \\ $$$${a}^{\mathrm{2}} +{t}^{\mathrm{2}} =\mathrm{18}.\:{find}\:{t}=? \\ $$ Answered by Ghisom_ last updated on 10/Mar/26 $$\forall{r}\in\mathbb{R}:\:\mathrm{fpart}\:\left({r}\right)\:={r}−\mathrm{ipart}\:\left({r}\right)…
Question Number 227983 by josemanuelrios last updated on 08/Mar/26 $$ \\ $$$$\sqrt{\mathrm{y}}+\mathrm{x}=\mathrm{a} \\ $$$$\sqrt{\mathrm{x}}+\mathrm{y}=\mathrm{a} \\ $$$$\forall\mathrm{x},\mathrm{y},\mathrm{a}\in\mathrm{Z} \\ $$$$ \\ $$ Answered by mr W last…
Question Number 227964 by mr W last updated on 05/Mar/26 Answered by Lara2440 last updated on 07/Mar/26 $$\mathrm{Let}'\mathrm{s}\:\mathrm{define}\:\mathrm{Sequence}\:{A}_{{n}} \:\mathrm{as}\:\frac{\left({n}!\right)^{\frac{\mathrm{1}}{{n}}} }{{n}}\:,\:{n}=\mathrm{1},\mathrm{2},\mathrm{3}….. \\ $$$$\mathrm{for}\:\mathrm{all}\:{n}>\mathrm{1}\:\mathrm{and}\:\mathrm{for}\:\mathrm{all}\:{x}>\mathrm{0} \\ $$$$\mathrm{1}+{nx}<\left({x}+\mathrm{1}\right)^{{n}} \:\centerdot\centerdot\centerdot\centerdot\centerdot\left(\mathrm{1}\right)\:\:\left(\mathrm{Bernoulli}\:\mathrm{Inequality}\right)…
Question Number 227954 by hardmath last updated on 04/Mar/26 $$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{holds}: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{6r}\:\leqslant\:\sqrt[{\mathrm{3}}]{\left(\mathrm{h}_{\boldsymbol{\mathrm{a}}} \:+\:\mathrm{h}_{\boldsymbol{\mathrm{b}}} \right)\centerdot\left(\mathrm{h}_{\boldsymbol{\mathrm{b}}} \:+\:\mathrm{h}_{\boldsymbol{\mathrm{c}}} \right)\centerdot\left(\mathrm{h}_{\boldsymbol{\mathrm{c}}} +\:\mathrm{h}_{\boldsymbol{\mathrm{a}}} \right)}\:\leqslant\:\mathrm{3R} \\ $$ Terms of Service Privacy Policy…
Question Number 227891 by mr W last updated on 24/Feb/26 Answered by Lara2440 last updated on 26/Feb/26 $${a}_{{n}+\mathrm{1}} =\sqrt{{a}_{{n}} +\mathrm{2}\:}\:\: \\ $$$$\: \\ $$$$ \\…
Question Number 227880 by hardmath last updated on 23/Feb/26 $$\mathrm{Find}:\:\:\:\frac{\mathrm{log}_{\mathrm{2}} ^{\mathrm{2}} \:\mathrm{20}\:−\:\mathrm{log}_{\mathrm{2}} ^{\mathrm{2}} \:\mathrm{5}}{\mathrm{log}_{\mathrm{2}} \mathrm{10}}\:=\:? \\ $$ Answered by efronzo1 last updated on 23/Feb/26 $$\:=\:\frac{\left(\mathrm{log}_{\mathrm{2}}…
Question Number 227878 by hardmath last updated on 23/Feb/26 $$\mathrm{Find}: \\ $$$$\mathrm{5}^{\left(\boldsymbol{\mathrm{log}}_{\mathrm{5}} \mathrm{3}\right)^{\mathrm{144}} } \:\:=\:\:? \\ $$ Commented by mr W last updated on 23/Feb/26…
Question Number 227853 by MrAjder last updated on 22/Feb/26 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{prove}:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{ln}\frac{{i}+\mathrm{1}}{{i}}\right)^{\mathrm{2}} <\frac{{n}}{{n}+\mathrm{1}} \\ $$ Answered by MrAjder last updated on 22/Feb/26 $$\forall{n}\in\mathbb{N}^{+} ,{S}_{{n}} =\underset{{i}=\mathrm{1}}…