Question Number 221352 by Nicholas666 last updated on 31/May/25 $$ \\ $$$$\:\:\:\:\mathrm{Given}\:\mathrm{real}\:\mathrm{numbers}\:{a},{b},{c}\:>\:\mathrm{0}\:, \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:{a}\:+\:{b}\:+\:{c}\:=\:{a}^{\mathrm{3}} \:+\:{b}^{\mathrm{3}} \:+\:{c}^{\mathrm{3}} \:, \\ $$$$\:\mathrm{Prove}\:;\:\frac{{a}^{\mathrm{3}} }{{a}^{\mathrm{4}} \:+\:{b}\:+\:{c}}\:+\:\frac{{b}^{\mathrm{3}} }{{b}^{\mathrm{4}} \:+\:{c}\:+\:{a}}\:+\:\frac{{c}^{\mathrm{3}} }{{c}^{\mathrm{4}} \:+\:\:{a}\:+\:{b}}\:\leqslant\:\mathrm{1}…
Question Number 221354 by Nicholas666 last updated on 31/May/25 $$ \\ $$$$\:\:\mathrm{Let}\:{a},{b},{c}\:\mathrm{be}\:\mathrm{there}\:\mathrm{real}\:\mathrm{numbers}, \\ $$$$\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}; \\ $$$$\:\mathrm{sin}\:{a}\:+\:\mathrm{sin}\:{b}\:+\:\mathrm{sin}\:{c}\:\geqslant\:\mathrm{2}\:\:\Rightarrow\:\mathrm{cos}\:{a}\:+\:\mathrm{cos}\:{b}\:+\:\mathrm{cos}\:{c}\:\leqslant\:\sqrt{\mathrm{5}}\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{and}, \\ $$$$\:\mathrm{sin}\:{a}\:+\:\mathrm{sin}\:{b}\:+\:\mathrm{sin}\:{c}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}}\:\Rightarrow\:\mathrm{cos}\left({a}−\pi/\mathrm{6}\right)\:+\:\mathrm{cos}\left({b}−\pi/\mathrm{6}\right)\:+\:\mathrm{cos}\left({c}−\pi/\mathrm{6}\right)\:\geqslant\:\mathrm{0}\:.\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$ Commented…
Question Number 221306 by Gbenga last updated on 30/May/25 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{csch}^{\mathrm{2}} \left(\pi{n}\right)}{{n}^{\mathrm{2}} } \\ $$ Answered by SdC355 last updated on 30/May/25 $$\underset{{l}=\mathrm{1}} {\overset{\infty}…
Question Number 221315 by klipto last updated on 30/May/25 $$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{function}}\:\boldsymbol{\mathrm{z}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{analytic}}\:\boldsymbol{\mathrm{within}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{simple}} \\ $$$$\boldsymbol{\mathrm{closed}}\:\boldsymbol{\mathrm{curve}}\:\boldsymbol{\mathrm{C}},−\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{z}}_{\mathrm{0}} \:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{point}}\:\boldsymbol{\mathrm{within}}\:\boldsymbol{\mathrm{C}} \\ $$$$\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{cauchy}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{integral}}\:\boldsymbol{\mathrm{formula}} \\ $$$$\oint\frac{\boldsymbol{\mathrm{sin}\pi\mathrm{z}}^{\mathrm{2}} +\boldsymbol{\mathrm{cos}\pi\mathrm{z}}^{\mathrm{2}} }{\left(\boldsymbol{\mathrm{x}}−\mathrm{1}\right)\left(\boldsymbol{\mathrm{x}}−\mathrm{2}\right)}\boldsymbol{\mathrm{dz}} \\ $$ Commented by MathematicalUser2357 last…
Question Number 221270 by SdC355 last updated on 29/May/25 $${p},{q}\in\mathbb{P}\: \\ $$$$\: \\ $$$$\mathrm{Use}\:\mathrm{prime}\:\mathrm{number}\:{p},{q}\:\mathrm{to}\:\mathrm{find}\:\mathrm{all}\:\mathrm{prime}\:\mathrm{number}\: \\ $$$$\mathrm{represented}\:\mathrm{by}\:{p}^{{q}} +{q}^{{p}} \\ $$ Answered by Frix last updated on…
Question Number 221233 by Nicholas666 last updated on 28/May/25 $$ \\ $$$$\:\mathrm{if}\:{a},{b},{c},{d},{e},{f}\:>\:\mathrm{0}\:\mathrm{and}\:{abcdef}\:=\:\mathrm{1}\:, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{then} \\ $$$$\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{ad}}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{be}}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{cf}}}\:\leqslant\:\frac{\mathrm{3}}{\:\sqrt{\mathrm{2}}}\: \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Profosed}\:\mathrm{by}\:\mathrm{Craciun}\:\mathrm{Georghe} \\ $$ Commented by Rasheed.Sindhi…
Question Number 221234 by Nicholas666 last updated on 28/May/25 $$ \\ $$$$\:\:\mathrm{let}\:\mathrm{0}\:\leqslant\:{a},{b},{c},\:\leqslant\:\mathrm{2}\:,\:\mathrm{and}\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\frac{\mathrm{3}}{\mathrm{2}}\:\leqslant\:\frac{\mathrm{1}}{{a}\:+\:\mathrm{1}}\:+\:\frac{\mathrm{1}}{{b}\:+\:\mathrm{1}}\:+\:\frac{\mathrm{1}}{{c}\:+\mathrm{1}\:}\:\leqslant\:\frac{\mathrm{11}}{\mathrm{6}} \\ $$$$ \\ $$ Commented by Frix last updated…
Question Number 221255 by MATHEMATICSAM last updated on 28/May/25 $$\mathrm{Let}\:\mathrm{X}\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{inside}\:\mathrm{a}\:\mathrm{square}\:\mathrm{ABCD},\: \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{XA}\:=\:\mathrm{10}\:\mathrm{cm},\:\mathrm{XB}\:=\:\mathrm{6}\:\mathrm{cm}\:\mathrm{and} \\ $$$$\mathrm{XC}\:=\:\mathrm{14}\:\mathrm{cm}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}. \\ $$ Commented by mr W last updated on 28/May/25 $${see}\:{also}\:{Q}\mathrm{72294}…
Question Number 221112 by MrGaster last updated on 25/May/25 Commented by SdC355 last updated on 25/May/25 $$\mathrm{oh}\:\mathrm{Sorry}….\:\mathrm{i}\:\mathrm{uploaded}\:\mathrm{wrong}…… \\ $$ Commented by MrGaster last updated on…
Question Number 221129 by SdC355 last updated on 25/May/25 $$\mathrm{prove} \\ $$$$\mathrm{Contour}\:\mathrm{integral}\:\mathrm{repreasentation} \\ $$$$\begin{pmatrix}{{p}}\\{{q}}\end{pmatrix}=\frac{\mathrm{1}}{\mathrm{2}\pi\boldsymbol{{i}}}\:\oint_{\:{C}} \:\left(\mathrm{1}−{z}\right)^{{p}} {z}^{−{q}} \:\frac{\mathrm{d}{z}}{{z}} \\ $$ Answered by MrGaster last updated on…