Question Number 217764 by ArshadS last updated on 20/Mar/25 $$ \\ $$$$\mathrm{Let}\:\mathrm{a},\:\mathrm{b},\:\mathrm{c}\:\mathrm{be}\:\mathrm{distinct}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\frac{{a}}{{b}−{c}}+\frac{{b}}{{c}−{a}}+\frac{{c}}{{a}−{b}}=\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{{a}}{\left({b}−{c}\right)^{\mathrm{2}} }+\frac{{b}}{\left({c}−{a}\right)^{\mathrm{2}} }+\frac{{c}}{\left({a}−{b}\right)^{\mathrm{2}} }=\mathrm{0} \\ $$ Answered by…
Question Number 217766 by ArshadS last updated on 20/Mar/25 $${Solve}\:{for}\:{x}\:\&\:{y} \\ $$$$\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}=\mathrm{5} \\ $$$$\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{{y}^{\mathrm{2}} }=\mathrm{13} \\ $$ Answered by Ghisom last updated on 20/Mar/25…
Question Number 217732 by ArshadS last updated on 19/Mar/25 $$\begin{cases}{{x}+{y}={xy}}\\{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{25}}\end{cases};\:\:{x}^{\mathrm{4}} +{y}^{\mathrm{4}} =? \\ $$ Commented by Ghisom last updated on 19/Mar/25 $${x}^{\mathrm{4}} +{y}^{\mathrm{4}}…
Question Number 217733 by Samuel12 last updated on 19/Mar/25 Answered by vnm last updated on 19/Mar/25 $${x}^{\frac{\mathrm{1}}{\mathrm{ln}\:\left({e}^{{x}} −\mathrm{1}\right)}} ={e}^{\frac{\mathrm{ln}\:{x}}{\mathrm{ln}\:\left({e}^{{x}} −\mathrm{1}\right)}} ={e}^{\frac{\mathrm{ln}\:{x}}{\mathrm{ln}\:{x}+\mathrm{ln}\:\frac{{e}^{{x}} −\mathrm{1}}{{x}}}} = \\ $$$${e}^{\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{ln}\:{x}}\mathrm{ln}\:\frac{{e}^{{x}}…
Question Number 217735 by ArshadS last updated on 19/Mar/25 $$\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}=\:\mathrm{abc} \\ $$$$\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:=\:\mathrm{49}\: \\ $$$$\mathrm{ab}+\mathrm{bc}+\mathrm{ca}=?\:\:\:.\:\:\:\:\:\:\:\:\:\:\:.\:\:\:\:\:\:\:\: \\ $$ Commented by mr W last updated…
Question Number 217730 by ArshadS last updated on 19/Mar/25 $${x}+\frac{\mathrm{1}}{{x}}=\mathrm{3}\:;\:{x}^{\mathrm{5}} +\frac{\mathrm{1}}{{x}^{\mathrm{5}} }=? \\ $$ Answered by Wuji last updated on 19/Mar/25 $${let}\:{S}_{{n}} ={x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }…
Question Number 217690 by hardmath last updated on 18/Mar/25 Commented by hardmath last updated on 18/Mar/25 $$\bigstar\mathrm{Hard}\bigstar \\ $$ Commented by mr W last updated…
Question Number 217660 by ArshadS last updated on 17/Mar/25 $$\:{f}\left({x}\right)\:+\:{f}\left({y}\right)={f}\left({x}+{y}\right)+{xy}\: \\ $$$${f}\left({x}\right)=? \\ $$ Answered by vnm last updated on 19/Mar/25 $${f}\left({x}\right)+{f}\left(\mathrm{0}\right)={f}\left({x}+\mathrm{0}\right)+{x}\centerdot\mathrm{0}\:\Rightarrow\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}\left({x}\right)+{f}\left(−{x}\right)={f}\left({x}−{x}\right)+{x}\left(−{x}\right)=−{x}^{\mathrm{2}} \\…
Question Number 217659 by ArshadS last updated on 17/Mar/25 $${x}+{y}=\mathrm{7}\:\wedge\:{x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\mathrm{133};\:{x},{y}=? \\ $$ Answered by Frix last updated on 17/Mar/25 $$\mathrm{2}+\mathrm{5}=\mathrm{7} \\ $$$$\mathrm{8}+\mathrm{125}=\mathrm{133} \\…
Question Number 217643 by Rojarani last updated on 17/Mar/25 $$\:{a}^{\frac{\mathrm{1}}{\mathrm{3}}} +{b}^{\frac{\mathrm{1}}{\mathrm{3}}} +{c}^{\frac{\mathrm{1}}{\mathrm{3}}} =\frac{\mathrm{1}}{\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{3}}} } \\ $$$$\:\:{b}+{c}−{a}=? \\ $$ Answered by mehdee7396 last updated on…