Question Number 142570 by mathsuji last updated on 02/Jun/21 $${Find}\:{all}\:{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:{such}\:{that} \\ $$$${f}\left({x}+{y}\right)=\mathrm{2}{f}\left({x}\right)+\mathrm{3}{f}\left({y}\right)−\mathrm{4}{xyf}\left(\mathrm{2}{x}−\mathrm{3}{y}\right) \\ $$$$\left(\forall{x};{y}\in\mathbb{R}\right) \\ $$ Answered by ajfour last updated on 02/Jun/21 $${f}\left({x}\right)=\mathrm{5}{f}\left(\frac{{x}}{\mathrm{2}}\right)−{x}^{\mathrm{2}} {f}\left(−\frac{{x}}{\mathrm{2}}\right)…
Question Number 77035 by necxxx last updated on 02/Jan/20 $${Solve}\:{for}\:{x}\:{in}: \\ $$$$\left({i}\right)\:\left(\mathrm{2}\left({x}+\mathrm{3}\right)−\mathrm{3}\left({x}−\mathrm{2}\right)\right)\left(\mathrm{2}{x}−\mathrm{1}\right)\geqslant\mathrm{0} \\ $$$$\left({ii}\right)\left({x}−\mathrm{1}\right)\left(\mathrm{2}{x}+\mathrm{3}\right)\left({x}+\mathrm{1}\right)\left({x}+\mathrm{3}\right)\leqslant\mathrm{1} \\ $$ Answered by MJS last updated on 03/Jan/20 $$\left({i}\right) \\…
Question Number 77028 by TawaTawa last updated on 02/Jan/20 Answered by mind is power last updated on 02/Jan/20 $$\mathrm{39} \\ $$$$\mathrm{x}−\mathrm{y}\:−\mathrm{z}\left(\mathrm{x}−\mathrm{y}\right)=\mathrm{0} \\ $$$$\mathrm{y}−\mathrm{z}−\mathrm{x}\left(\mathrm{y}−\mathrm{z}\right)=\mathrm{0} \\ $$$$\mathrm{x}^{\mathrm{2}}…
Question Number 77033 by necxxx last updated on 02/Jan/20 $${Suppose}\:{the}\:{population}\:{models}\:{of}\:{London} \\ $$$${and}\:{Hongkong}\:{in}\:{tens}\:{of}\:{thousands}\:{are} \\ $$$${p}\left({t}\right)=\frac{\mathrm{20}{t}}{{t}+\mathrm{1}}\:{and}\:{q}\left({t}\right)=\frac{\mathrm{240}{t}}{{t}+\mathrm{8}}\:{respectively}\:{for} \\ $$$${t}\:{years}\:{after}\:\mathrm{2015},\:{Determine}\:{the}\:{time}\:{period} \\ $$$${in}\:{years}\:{when}\:{the}\:{population}\:{of}\:{London} \\ $$$${exceeds}\:{that}\:{of}\:{HongKong}. \\ $$ Commented by MJS…
Question Number 142556 by iloveisrael last updated on 02/Jun/21 $$\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2}{yz}+\mathrm{2}−{z}^{\mathrm{2}} }\\{{z}=\mathrm{4022}−{x}−{y}}\end{cases} \\ $$$$\:{x},{y},{z}\in\mathbb{Z}^{+} \:.\:{then}\:{z}= \\ $$ Answered by iloveisrael last updated on 02/Jun/21…
Question Number 77024 by Sathvik last updated on 02/Jan/20 $$\mathrm{The}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:\mathrm{p}\:\mathrm{for}\:\mathrm{which}\: \\ $$$$\mathrm{graph}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{2p}^{\mathrm{2}} − \\ $$$$\mathrm{3ptan}\:\mathrm{x}+\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}+\mathrm{1}\:\mathrm{does}\:\mathrm{not}\:\mathrm{lie}\:\mathrm{below}\: \\ $$$$\mathrm{x}-\mathrm{axis}\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}\in\left(\frac{−\Pi}{\mathrm{2}},\frac{\Pi}{\mathrm{2}}\right)\:\mathrm{is} \\ $$$$\left(\mathrm{a}\right)\mathrm{0}\:\:\:\:\:\:\left(\mathrm{b}\right)\mathrm{4}\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\mathrm{3}\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\mathrm{8} \\ $$ Answered by MJS…
Question Number 142548 by ivoo last updated on 02/Jun/21 Commented by ivoo last updated on 02/Jun/21 How many integer solutions? Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 77014 by aliesam last updated on 02/Jan/20 Commented by mr W last updated on 02/Jan/20 $${look}\:{and}\:{think}: \\ $$$${x}={y}=\pm\mathrm{1} \\ $$ Commented by mr…
Question Number 11476 by tawa last updated on 26/Mar/17 Answered by sandy_suhendra last updated on 27/Mar/17 $$\frac{\mathrm{1}}{\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{b}−\mathrm{c}\right)}\:+\:\frac{\mathrm{1}}{\left(\mathrm{b}−\mathrm{c}\right)\left(\mathrm{c}−\mathrm{a}\right)}\:+\:\frac{\mathrm{1}}{\left(\mathrm{c}−\mathrm{a}\right)\left(\mathrm{a}−\mathrm{b}\right)}\:\: \\ $$$$=\frac{\mathrm{c}−\mathrm{a}+\mathrm{a}−\mathrm{b}+\mathrm{b}−\mathrm{c}}{\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{b}−\mathrm{c}\right)\left(\mathrm{c}−\mathrm{a}\right)}\:=\:\mathrm{0} \\ $$$$ \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{a}−\mathrm{b}\right)^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{b}−\mathrm{c}\right)^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{c}−\mathrm{a}\right)^{\mathrm{2}}…
Question Number 11477 by @ANTARES_VY last updated on 26/Mar/17 $$\frac{\boldsymbol{\mathrm{a}}^{\mathrm{8}} +\boldsymbol{\mathrm{a}}^{\mathrm{4}} +\mathrm{1}}{\boldsymbol{\mathrm{a}}^{\mathrm{6}} +\mathrm{1}}. \\ $$$$\boldsymbol{\mathrm{solves}}…. \\ $$ Answered by sm3l2996 last updated on 26/Mar/17 $$\mathrm{a}^{\mathrm{8}}…