Question Number 3465 by Rasheed Soomro last updated on 13/Dec/15 $${Find}\:{the}\:{value}\:{of}\:{n}\:{so}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\frac{{a}^{{n}+\mathrm{1}} +{b}^{{n}+\mathrm{1}} }{{a}^{{n}} +{b}^{{n}} } \\ $$$${may}\:{become}\:{the}\:{G}.{M}.\:{between}\: \\ $$$${a}\:\:{and}\:\:\:{b}. \\ $$ Answered by…
Question Number 3466 by Rasheed Soomro last updated on 13/Dec/15 $${Find}\:{the}\:{value}\:{of}\:{n}\:{so}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{{n}+\mathrm{1}} +{b}^{{n}+\mathrm{1}} }{{a}^{{n}} +{b}^{{n}} } \\ $$$${may}\:{become}\:\:{the}\:{A}.{M}.\:{between} \\ $$$${a}\:\:{and}\:\:{b}. \\ $$ Answered by…
Question Number 3464 by Rasheed Soomro last updated on 13/Dec/15 $$\mathcal{I}{f}\:\:{a},{b},{c},{d}\:{are}\:{in}\:{G}.{P}.,\:{prove}\:{that} \\ $$$${a}^{\mathrm{2}} −{b}^{\mathrm{2}} ,{b}^{\mathrm{2}} −{c}^{\mathrm{2}} ,{c}^{\mathrm{2}} −{d}^{\mathrm{2}} \:{are}\:{also}\:{in}\:{G}.{P}. \\ $$ Answered by Yozzii last…
Question Number 3451 by prakash jain last updated on 13/Dec/15 $$\mathrm{Consider}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{equation} \\ $$$$\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{i}} {x}^{{i}} =\mathrm{0},\:{a}_{{i}} \in\mathbb{Z} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{a}+{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$$$\mathrm{equation}\:\mathrm{then}\:{a}−{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{a}\:\mathrm{root}. \\ $$$${a},{b},{c}\in\mathbb{Z},\:{c}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{whole}\:\mathrm{square}. \\…
Question Number 68960 by ajfour last updated on 17/Sep/19 Commented by TawaTawa last updated on 17/Sep/19 $$\mathrm{Wow},\:\mathrm{weldone}\:\mathrm{sir},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$ Commented by ajfour last updated on…
Question Number 68935 by Maclaurin Stickker last updated on 17/Sep/19 $${Find}\:{all}\:{values}\:{for}\:\boldsymbol{{x}}: \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{7}{x}+\mathrm{11}\right)^{{x}^{\mathrm{2}} −\mathrm{13}{x}+\mathrm{42}} =\mathrm{1} \\ $$$$\left({Easy}\right) \\ $$ Answered by $@ty@m123 last updated…
Question Number 134467 by bramlexs22 last updated on 04/Mar/21 $$\begin{array}{|c|c|}{\mathrm{If}\:\mathrm{x}^{\mathrm{2}} =\mathrm{2y}+\mathrm{5}\:\mathrm{and}\:\mathrm{y}^{\mathrm{2}} =\mathrm{2x}+\mathrm{5}\:}\\{\mathrm{what}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} −\mathrm{2x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} .}\\\hline\end{array} \\ $$ Answered by benjo_mathlover last updated on 04/Mar/21…
Question Number 68912 by TawaTawa last updated on 16/Sep/19 Answered by MJS last updated on 17/Sep/19 $$\left(\mathrm{1}\right)\:\:{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} −\mathrm{7}=\mathrm{0} \\ $$$$\left(\mathrm{2}\right)\:\:{b}^{\mathrm{2}} +{bc}+{c}^{\mathrm{2}} −\mathrm{21}=\mathrm{0} \\ $$$$\left(\mathrm{3}\right)\:\:{c}^{\mathrm{2}}…
Question Number 68898 by Rio Michael last updated on 16/Sep/19 $${solve}\:{for}\:{x}\:{the}\:{equation} \\ $$$$\:\:{log}_{{x}} {e}^{\mathrm{2}{x}} \:=\:{eln}\:{x}\:−{e} \\ $$ Commented by kaivan.ahmadi last updated on 16/Sep/19 $$\frac{\mathrm{2}{x}}{{lnx}}={elnx}−{e}\Rightarrow{e}\left({lnx}\right)^{\mathrm{2}}…
Question Number 68899 by Rio Michael last updated on 16/Sep/19 $${solve}\:{for}\:{x}\:{and}\:{y}\:{the}\:{equation} \\ $$$$\:\mathrm{2}{lnx}\:−{lny}\:={ln}\left(\mathrm{5}{x}−\mathrm{6}{y}\right) \\ $$ Commented by kaivan.ahmadi last updated on 16/Sep/19 $${i}\:{dont}\:{know}.{i}\:{think}\:{it}\:{is}\:{hard}. \\ $$…