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Category: Algebra

f-f-x-1-1-x-2-f-x-

Question Number 171612 by mathlove last updated on 18/Jun/22 $${f}\left({f}\left({x}\right)\right)=\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:{f}\left({x}\right)=? \\ $$ Commented by mr W last updated on 19/Jun/22 $${i}\:{don}'{t}\:{think}\:{such}\:{a}\:{function}\:{equation} \\ $$$${is}\:{solvable}.\:{if}\:{you}\:{have}\:{a}\:{solution}, \\…

if-f-f-x-x-2-x-1-find-f-0-

Question Number 171583 by mr W last updated on 18/Jun/22 $${if}\:{f}\left({f}\left({x}\right)\right)={x}^{\mathrm{2}} −{x}+\mathrm{1},\:{find}\:{f}\left(\mathrm{0}\right)=? \\ $$ Commented by infinityaction last updated on 18/Jun/22 $${f}\left({f}\left({f}\left({x}\right)\right)\right)\:=\:{f}\left({x}\right)^{\mathrm{2}} −{f}\left({x}\right)+\mathrm{1} \\ $$$$\:\:\:{f}\left({f}\left(\mathrm{0}\right)\right)\:=\:\mathrm{1}…

Show-that-4-3n-2-5-is-divisible-by-9-then-simplify-4-3n-2-5-9-

Question Number 171572 by Raxreedoroid last updated on 18/Jun/22 $$\mathrm{Show}\:\mathrm{that} \\ $$$$\mathrm{4}^{\mathrm{3}{n}−\mathrm{2}} +\mathrm{5}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{9} \\ $$$$\mathrm{then}\:\mathrm{simplify}\:\frac{\mathrm{4}^{\mathrm{3}{n}−\mathrm{2}} +\mathrm{5}}{\mathrm{9}} \\ $$ Commented by kaivan.ahmadi last updated on 18/Jun/22…

evaluate-a-b-7-b-c-7-c-a-7-a-b-3-b-c-3-c-a-3-a-2-b-2-c-2-ab-bc-ca-

Question Number 171563 by infinityaction last updated on 19/Jun/22 $$\:\:\:\:\:\:\:\:{evaluate}\:\:\: \\ $$$$\:\:\:\:\:\frac{\sqrt{\frac{\left({a}−{b}\right)^{\mathrm{7}} \:+\:\left({b}−{c}\right)^{\mathrm{7}} \:+\:\left({c}−{a}\right)^{\mathrm{7}} }{\left({a}−{b}\right)^{\mathrm{3}} \:+\:\left({b}−{c}\right)^{\mathrm{3}} \:+\:\left({c}−{a}\right)^{\mathrm{3}} }}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}}\:=\:\:?? \\ $$ Answered by…

In-ABC-I-incenter-ID-BC-IE-CA-IF-AB-D-BC-E-CA-F-AB-I-a-I-b-I-c-excenters-Prove-that-cyc-EF-sin-A-2-cyc-EF-sin-A-2-1-4r-2-R-I-a-I-

Question Number 171558 by Shrinava last updated on 17/Jun/22 $$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:,\:\mathrm{I}-\mathrm{incenter} \\ $$$$\mathrm{ID}\bot\mathrm{BC}\:,\:\mathrm{IE}\bot\mathrm{CA}\:,\:\mathrm{IF}\bot\mathrm{AB} \\ $$$$\mathrm{D}\in\left(\mathrm{BC}\right)\:,\:\mathrm{E}\in\left(\mathrm{CA}\right)\:,\:\mathrm{F}\in\left(\mathrm{AB}\right) \\ $$$$\mathrm{I}_{\boldsymbol{\mathrm{a}}} \:,\:\mathrm{I}_{\boldsymbol{\mathrm{b}}} \:,\:\mathrm{I}_{\boldsymbol{\mathrm{c}}} -\mathrm{excenters}.\:\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{EF}}{\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}}\:\:+\:\:\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\frac{\mathrm{EF}}{\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}}\:\:=\:\:\frac{\mathrm{1}\:+\:\mathrm{4}\boldsymbol{\mathrm{r}}^{\mathrm{2}} }{\mathrm{R}}\:\centerdot\:\left[\mathrm{I}_{\boldsymbol{\mathrm{a}}} \mathrm{I}_{\boldsymbol{\mathrm{b}}}…

Solve-for-real-numbers-2x-2-3y-2-z-2-7-x-2-y-2-z-2-2-z-x-y-

Question Number 171549 by Shrinava last updated on 17/Jun/22 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\begin{cases}{\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{3y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{7}}\\{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\sqrt{\mathrm{2}}\:\mathrm{z}\:\left(\mathrm{x}\:+\:\mathrm{y}\right)}\end{cases} \\ $$ Answered by MJS_new last updated…