Question Number 194695 by horsebrand11 last updated on 13/Jul/23 $$\:\:\:\:\:\downharpoonleft\underline{\:} \\ $$ Answered by som(math1967) last updated on 13/Jul/23 $$\boldsymbol{{let}}\:\frac{\boldsymbol{{x}}}{\boldsymbol{{a}}+\boldsymbol{{b}}−\boldsymbol{{c}}}=\frac{\boldsymbol{{y}}}{\boldsymbol{{b}}+\boldsymbol{{c}}−\boldsymbol{{a}}}=\frac{\boldsymbol{{z}}}{\boldsymbol{{c}}+\boldsymbol{{a}}−\boldsymbol{{b}}}=\boldsymbol{{k}} \\ $$$$\boldsymbol{{x}}=\boldsymbol{{k}}\left(\boldsymbol{{a}}+\boldsymbol{{b}}−\boldsymbol{{c}}\right) \\ $$$$\boldsymbol{{y}}=\boldsymbol{{k}}\left(\boldsymbol{{b}}+\boldsymbol{{c}}−\boldsymbol{{a}}\right) \\…
Question Number 194689 by cortano12 last updated on 13/Jul/23 Answered by MM42 last updated on 13/Jul/23 $${f}_{{i}} :\mathrm{1},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8},\mathrm{13},\mathrm{21},\mathrm{34},\mathrm{55},\mathrm{89},… \\ $$$$\Rightarrow\forall\:{n}=\mathrm{5}{k}\:\:;\:\mathrm{5}\mid{f}_{{n}} \:\:\:!!?? \\ $$$$\Rightarrow{if}\:\:{n}<\mathrm{10000}\:\rightarrow\:{ans}=\mathrm{4999}\:\checkmark \\ $$$${if}\:\:{f}_{{n}}…
Question Number 194688 by CrispyXYZ last updated on 13/Jul/23 $$\mathrm{find}\:\mathrm{all}\:\mathrm{function}\:{f}:\:\mathbb{R}\:\rightarrow\:\mathbb{R}\:\mathrm{such}\:\mathrm{that}\:\forall{x},\:{y}\in\mathbb{R}, \\ $$$${f}\left({x}−{f}\left({y}\right)\right)={f}\left({f}\left({y}\right)\right)+{xf}\left({y}\right)+{f}\left({x}\right)−\mathrm{1}. \\ $$ Answered by Tinku Tara last updated on 13/Jul/23 $${put}\:{x}={f}\left({y}\right) \\ $$$${f}\left(\mathrm{0}\right)={f}\left({x}\right)+{x}^{\mathrm{2}}…
Question Number 194691 by CrispyXYZ last updated on 13/Jul/23 $${a},\:{b},\:{c}\geqslant\mathrm{0},\:{a}+{b}+{c}=\mathrm{2}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{3}{a}+\mathrm{8}{ab}+\mathrm{16}{abc}\leqslant\mathrm{12}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
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Question Number 194654 by sonukgindia last updated on 12/Jul/23 Answered by MM42 last updated on 12/Jul/23 $$\left(\left(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{{tanx}} \right)^{{tanx}+\mathrm{2}} ={u} \\ $$$$\Rightarrow{u}+\frac{\mathrm{1}}{{u}}=\mathrm{6}\Rightarrow{u}^{\mathrm{2}} −\mathrm{6}{u}+\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow{u}=\mathrm{3}\pm\mathrm{2}\sqrt{\mathrm{2}} \\…
Question Number 194649 by SANOGO last updated on 12/Jul/23 $${calcul}\: \\ $$$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}+{cos}^{\mathrm{2}} {t}\:}{dt} \\ $$ Commented by Frix last updated on 12/Jul/23…
Question Number 194648 by universe last updated on 12/Jul/23 Commented by Frix last updated on 12/Jul/23 $$\frac{\mathrm{7}}{\mathrm{8}} \\ $$ Commented by universe last updated on…
Question Number 194640 by cortano12 last updated on 12/Jul/23 $$\:\:\:\:\underbrace{ } \\ $$ Answered by horsebrand11 last updated on 12/Jul/23 $$\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}}}\:=\:\mathrm{y} \\ $$$$\:\underset{\mathrm{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}}{\mathrm{y}\:\mathrm{sin}\:\mathrm{3y}}\:=\underset{\mathrm{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}}…
Question Number 194642 by horsebrand11 last updated on 12/Jul/23 $$\:\mathrm{If}\:\mathrm{A}=\begin{pmatrix}{\mathrm{a}\:\:\:\:\mathrm{b}\:\:\:\:\:\:\mathrm{c}}\\{\mathrm{b}\:\:\:\:\mathrm{c}\:\:\:\:\:\:\mathrm{a}}\\{\mathrm{c}\:\:\:\:\:\mathrm{a}\:\:\:\:\:\:\mathrm{b}}\end{pmatrix}\:\mathrm{and}\:\mathrm{a},\mathrm{b},\mathrm{c}\:>\mathrm{0} \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{abc}=\mathrm{1}\:\mathrm{and}\:\mathrm{A}^{\mathrm{T}} .\mathrm{A}=\mathrm{I} \\ $$$$\:\mathrm{find}\:\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} −\mathrm{3abc}\:. \\ $$ Answered by som(math1967) last updated…