Question Number 150887 by mathdanisur last updated on 16/Aug/21 Commented by dumitrel last updated on 16/Aug/21 $${a}\leqslant{b}\leqslant{c}\leqslant{d}\Rightarrow\left({a}+{b}+{c}+{d}\right)^{\mathrm{2}} \geqslant\mathrm{8}\left({ac}+{bd}\right) \\ $$ Commented by mathdanisur last updated…
Question Number 150881 by mathdanisur last updated on 16/Aug/21 $$\mathrm{y}\:=\:\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{\boldsymbol{\mathrm{x}}} \centerdot\:\left(\frac{\mathrm{b}}{\mathrm{x}}\right)^{\boldsymbol{\mathrm{a}}} \centerdot\:\left(\frac{\mathrm{x}}{\mathrm{a}}\right)^{\boldsymbol{\mathrm{b}}} \:\Rightarrow\:\mathrm{y}\:^{'} \:=\:? \\ $$ Answered by john_santu last updated on 16/Aug/21 $$\mathrm{y}=\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{\mathrm{x}} \left(\frac{\mathrm{x}}{\mathrm{b}}\right)^{\mathrm{b}−\mathrm{a}}…
Question Number 150880 by mathdanisur last updated on 16/Aug/21 $$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{1} \\ $$$$\mathrm{find}\:\boldsymbol{\mathrm{min}}\left(\frac{\mathrm{1}}{\mathrm{a}}\:+\:\frac{\mathrm{9}}{\mathrm{b}}\:+\:\frac{\mathrm{25}}{\mathrm{c}}\right)\:=\:? \\ $$$$\left.\mathrm{a}\left.\right)\left.\mathrm{7}\left.\mathrm{3}\left.\:\:\:\mathrm{b}\right)\mathrm{75}\:\:\:\mathrm{c}\right)\mathrm{105}\:\:\:\mathrm{d}\right)\mathrm{81}\:\:\:\mathrm{e}\right)\mathrm{83} \\ $$ Answered by mr W last updated on 16/Aug/21 $${s}=\frac{\mathrm{1}}{{a}}+\frac{\mathrm{9}}{{b}}+\frac{\mathrm{25}}{\mathrm{1}−{a}−{b}}…
Question Number 150883 by mathdanisur last updated on 16/Aug/21 $$\mathrm{prove}\:\mathrm{that}\:\mathrm{any}\:\mathrm{real}\:\mathrm{root}\:\boldsymbol{\alpha}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}:\:\:\mathrm{x}^{\mathrm{6}\boldsymbol{\mathrm{n}}} \:=\:\mathrm{4x}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:+\:\mathrm{4}\:;\:\mathrm{n}\in\mathbb{N}-\left\{\mathrm{0}\right\} \\ $$$$\mathrm{verify}:\:\:\mid\boldsymbol{\alpha}\mid\:>\:\sqrt[{\mathrm{2}\boldsymbol{\mathrm{n}}}]{\mathrm{2}} \\ $$ Answered by mindispower last updated on 16/Aug/21…
Question Number 19799 by Tinkutara last updated on 15/Aug/17 $$\mathrm{For}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number}\:{b},\:\mathrm{let}\:{N}\left({b}\right)\:\mathrm{denote} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{numbers}\:{a}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:=\:\mathrm{0}\:\mathrm{has} \\ $$$$\mathrm{integer}\:\mathrm{roots}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{smallest} \\ $$$$\mathrm{value}\:\mathrm{of}\:{b}\:\mathrm{for}\:\mathrm{which}\:{N}\left({b}\right)\:=\:\mathrm{20}? \\ $$ Commented by dioph last…
Question Number 150870 by liberty last updated on 16/Aug/21 Commented by mnjuly1970 last updated on 16/Aug/21 $$\:\:{pls}\:\:{solve}\:{it}… \\ $$ Answered by tabata last updated on…
Question Number 150865 by Tawa11 last updated on 16/Aug/21 Im{f'(z)} = 6x(2y-1) and f(0) = 3 – 2i, f(1) = 6 – 5i. Find f(2…
Question Number 85322 by mathocean1 last updated on 20/Mar/20 $${please}\:{help}\:{me}\:{to}\:{solve}\:{this}\:{in}\:\mathbb{N} \\ $$$${A}_{{n}} ^{\mathrm{2}} −\mathrm{3}{C}_{{n}} ^{\:{n}−\mathrm{2}} +{n}=−\mathrm{20} \\ $$ Commented by mr W last updated on…
Question Number 19785 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:{x}^{\left({x}^{\mathrm{4}} \right)} \:=\:\mathrm{4},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${x}^{\left({x}^{\mathrm{2}} \right)} \:+\:{x}^{\left({x}^{\mathrm{8}} \right)} ? \\ $$ Answered by Tinkutara last updated…
Question Number 19786 by Tinkutara last updated on 15/Aug/17 $$\mathrm{For}\:\mathrm{natural}\:\mathrm{numbers}\:{x}\:\mathrm{and}\:{y},\:\mathrm{let}\:\left({x},\:{y}\right) \\ $$$$\mathrm{denote}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{common}\:\mathrm{divisor}\:\mathrm{of} \\ $$$${x}\:\mathrm{and}\:{y}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{natural} \\ $$$$\mathrm{numbers}\:{x}\:\mathrm{and}\:{y}\:\mathrm{with}\:{x}\:\leqslant\:{y}\:\mathrm{satisfy}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{xy}\:=\:{x}\:+\:{y}\:+\:\left({x},\:{y}\right)? \\ $$ Answered by mrW1 last updated…