Question Number 150571 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Find}\:\boldsymbol{\mathrm{A}}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{2021}\in\boldsymbol{\mathrm{A}}\:\mathrm{if} \\ $$$$\overline {\mathrm{abcd}}\in\boldsymbol{\mathrm{A}},\:\:\frac{\mathrm{a}}{\mathrm{d}\:+\:\mathrm{1}}\:=\:\frac{\mathrm{c}\:-\:\mathrm{b}}{\mathrm{c}}\:=\:\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{b}\:+\:\mathrm{c}} \\ $$ Answered by Rasheed.Sindhi last updated on 15/Aug/21 $$\overline {\mathrm{abcd}}\in\boldsymbol{\mathrm{A}};\:\:\frac{\mathrm{a}}{\mathrm{d}\:+\:\mathrm{1}}\:=\:\frac{\mathrm{c}\:-\:\mathrm{b}}{\mathrm{c}}\:=\:\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{b}\:+\:\mathrm{c}};\mathrm{A}=? \\…
Question Number 150559 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{natural}\:\mathrm{numbers}: \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{4}} \:=\:\mathrm{2022} \\ $$ Answered by MJS_new last updated on 13/Aug/21 $$\mathrm{if}\:{x}=\mathrm{0}:\:{y}^{\mathrm{4}} =\mathrm{2022}\:\Rightarrow\:{y}\approx\mathrm{6}.\mathrm{7}…
Question Number 150558 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{integers}: \\ $$$$\left(\mathrm{6x}\:+\:\mathrm{5y}^{\mathrm{2}} \right)\left(\mathrm{4z}\:+\:\mathrm{x}\right)\left(\mathrm{2y}^{\mathrm{2}} \:+\:\mathrm{3z}\right)\:=\:\mathrm{2021} \\ $$ Answered by MJS_new last updated on 13/Aug/21 $$\mathrm{2021}=\mathrm{43}×\mathrm{47} \\…
Question Number 150549 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Prove}\:\mathrm{that}:\:\:\forall\mathrm{n}\in\mathbb{N} \\ $$$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\prod}}\mathrm{k}!\:\centerdot\:\mathrm{k}^{\boldsymbol{\mathrm{n}}−\boldsymbol{\mathrm{k}}+\mathrm{1}} \:\leqslant\:\left(\frac{\mathrm{n}+\mathrm{2}}{\mathrm{3}}\right)^{\boldsymbol{\mathrm{n}}\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 150550 by bekzodjumayev last updated on 13/Aug/21 Commented by bekzodjumayev last updated on 13/Aug/21 $$\boldsymbol{{H}}{elp}?? \\ $$ Commented by liberty last updated on…
Question Number 150548 by mathdanisur last updated on 13/Aug/21 $$\mathrm{For}\:\:\boldsymbol{\mathrm{k}}<\mathbb{N}\:\:\mathrm{fixed}\:\:\mathrm{and}\:\:\boldsymbol{\alpha}>\mathrm{0}\:\:\mathrm{then}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}^{\boldsymbol{\alpha}} }}\:\centerdot\:\left(\frac{\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\prod}}\left(\mathrm{n}+\mathrm{k}+\mathrm{i}\right)}{\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\prod}}\left(\mathrm{n}+\mathrm{i}\right)}\right)^{\boldsymbol{\mathrm{n}}^{\boldsymbol{\alpha}} } \\ $$ Terms of Service Privacy…
Question Number 150539 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{the}\:\mathrm{foolowing}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\frac{\boldsymbol{\mathrm{n}}^{\mathrm{2}} +\boldsymbol{\mathrm{n}}+\mathrm{2}}{\mathrm{2}}} \:\mathrm{e}^{−\boldsymbol{\pi\mathrm{n}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:=\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{e}^{−\boldsymbol{\pi\mathrm{n}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \\ $$ Answered by…
Question Number 150531 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Find}\:\:\mathrm{x};\mathrm{y}\:\:;\:\:\mathrm{x}\in\mathrm{Q}\:\:\mathrm{and}\:\:\mathrm{y}\in\mathrm{Z}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\mathrm{2020}\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \right)\:+\:\mathrm{2019}\left(\mathrm{x}\:+\:\mathrm{y}\right)\:=\:\mathrm{2021xy} \\ $$ Commented by Rasheed.Sindhi last updated on 14/Aug/21 $$\underset{\smile} {\overset{\frown}…
Question Number 19455 by Tinkutara last updated on 11/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{z}\:=\:\mathrm{cos}\:\mathrm{6}°\:+\:{i}\:\mathrm{sin}\:\mathrm{6}°,\:\mathrm{then} \\ $$$$\frac{\mathrm{1}}{{z}^{\mathrm{2}} \:+\:\mathrm{1}}\:−\:\frac{{iz}}{{z}^{\mathrm{4}} \:−\:\mathrm{1}}\:+\:\frac{{iz}^{\mathrm{3}} }{{z}^{\mathrm{8}} \:−\:\mathrm{1}}\:+\:\frac{{iz}^{\mathrm{7}} }{{z}^{\mathrm{16}} \:−\:\mathrm{1}}\:=\:\mathrm{0}. \\ $$ Answered by ajfour last updated…
Question Number 150515 by mathdanisur last updated on 13/Aug/21 Answered by ajfour last updated on 13/Aug/21 $${x}={t}+{h} \\ $$$$\mathrm{7}\sqrt{\mathrm{4}{t}^{\mathrm{2}} +\mathrm{8}{ht}+\mathrm{4}{h}^{\mathrm{2}} +\mathrm{5}{t}+\mathrm{5}{h}−\mathrm{1}} \\ $$$$−\mathrm{7}\sqrt{\mathrm{4}{t}^{\mathrm{2}} +\mathrm{8}{ht}+\mathrm{4}{h}^{\mathrm{2}} −\mathrm{12}{t}−\mathrm{12}{h}+\mathrm{12}}…