Question Number 188270 by mnjuly1970 last updated on 27/Feb/23 Commented by mnjuly1970 last updated on 27/Feb/23 $$\:\:{in}\:\:{A}\overset{\Delta} {{B}C}\:{prove}\::\: \\ $$$$\:\:\:\frac{\mathrm{1}}{{cos}\left({A}\right)}\:+\frac{\mathrm{1}}{{cos}\left({B}\right)}\:+\frac{\mathrm{1}}{{cos}\left({C}\right)}\:\geqslant\:\mathrm{6} \\ $$$$\: \\ $$$$\:\:\:\:\:{note}\::\:\mathrm{0}\:<\:{A}\:,\:{B}\:,\:{C}\:<\:\mathrm{90}^{°} \\…
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Question Number 188262 by normans last updated on 27/Feb/23 $$ \\ $$$$\:\:\:\:\:\:\:\boldsymbol{{solve}}\:\boldsymbol{{the}}\:\boldsymbol{{equation}};\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\left.\begin{matrix}{\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:+\boldsymbol{{z}}\:=\:\:\mathrm{30}\sqrt{\mathrm{2}}}\\{\boldsymbol{{x}}\:−\:\boldsymbol{{y}}\:−\:\boldsymbol{{z}}\:=\:\mathrm{7},\mathrm{5}}\\{\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:−\:\boldsymbol{{z}}\:=\:\sqrt{\mathrm{22}}}\end{matrix}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}\:;\:\boldsymbol{{y}}\:;\:\boldsymbol{{z}}\:=\:?? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{they}\:{form}\:{funny}\:{positions}\: \\ $$$$ \\ $$ Answered…
Question Number 188248 by mathlove last updated on 27/Feb/23 $$\left({x}^{\mathrm{3}} −{y}−\mathrm{3}{x}\right)\left[\left({x}^{\mathrm{3}} −\mathrm{3}{x}\right)^{\mathrm{2}} −{y}^{\mathrm{2}} \right]=\mathrm{200} \\ $$$$\left({x}^{\mathrm{3}} +{y}−\mathrm{3}{x}\right)\left[\left({x}^{\mathrm{3}} −\mathrm{3}{x}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} \right]=\mathrm{600} \\ $$$${solved}\:{in}\:{R} \\ $$ Terms…
Question Number 188247 by cortano12 last updated on 27/Feb/23 $$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\left(\mathrm{1}\right)\:\mathrm{5555}^{\mathrm{2222}} +\mathrm{2222}^{\mathrm{5555}} \:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{3}^{\mathrm{105}} +\mathrm{4}^{\mathrm{105}} \:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}\: \\ $$ Answered by Rasheed.Sindhi last updated…
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Question Number 188224 by Shrinava last updated on 26/Feb/23 $$\mathrm{If}\:\:\:\Omega\:=\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\underset{\boldsymbol{\mathrm{k}}=\mathrm{2}} {\overset{\infty} {\prod}}\:\frac{\mathrm{k}^{\mathrm{3}} \:−\:\mathrm{1}}{\mathrm{k}^{\mathrm{3}} \:+\:\mathrm{1}}\right)^{\boldsymbol{\mathrm{n}}} \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{complex}\:\mathrm{numbees}: \\ $$$$\mathrm{z}^{\mathrm{4}} \:+\:\mathrm{3z}^{\mathrm{3}} \:+\:\Omega\mathrm{z}^{\mathrm{2}} \:+\:\mathrm{3z}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$…
Question Number 188226 by Shrinava last updated on 26/Feb/23 $$\mathrm{If}\:\:\:\Omega\:=\:\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{sin}\left(\mathrm{A}\:−\:\frac{\pi}{\mathrm{6}}\right)}{\mathrm{cos}\left(\mathrm{B}\:−\:\frac{\pi}{\mathrm{6}}\right)\mathrm{cos}\left(\mathrm{C}\:−\:\frac{\pi}{\mathrm{6}}\right)}\:\:\:\mathrm{in}\:\:\bigtriangleup\mathrm{ABC} \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{x}^{\mathrm{4}} \:−\:\mathrm{4}\Omega\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{6}\Omega\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{4}\Omega\mathrm{x}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$ Answered by aleks041103 last updated…
Question Number 57127 by ANTARES VY last updated on 30/Mar/19 $$\left\{\boldsymbol{\mathrm{cos}}\mathrm{1}°\right\}+\left\{\boldsymbol{\mathrm{cos}}\mathrm{2}°\right\}+\left\{\boldsymbol{\mathrm{cos}}\mathrm{3}°\right\}+….+\left\{\boldsymbol{\mathrm{cos}}\mathrm{270}\right\}=? \\ $$ Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19 $$\boldsymbol{{source}}\:\boldsymbol{{S}}.\boldsymbol{{L}}\:\boldsymbol{{Loney}}\:\boldsymbol{{great}}\:\boldsymbol{{mathematician}} \\ $$ Commented by…
Question Number 122658 by mathocean1 last updated on 18/Nov/20 $${we}\:{are}\:{in}\:\mathbb{C}^{\mathrm{3}} . \\ $$$$\left({S}\right):\:\:\begin{cases}{{x}+{y}+{z}=\mathrm{2}{i}−\mathrm{1}\:{and}\:{xyz}=\mathrm{2}}\\{{xy}+{yz}+{xz}=−\mathrm{2}\left(\mathrm{1}+{i}\right)\:}\end{cases} \\ $$$${P}\left({z}\right)={z}^{\mathrm{3}} +\left(\mathrm{1}−\mathrm{2}{i}\right){z}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{1}+{i}\right)−\mathrm{2}. \\ $$$${show}\:{that}\:\left({a},{b},{c}\right)\:{is}\:{solution} \\ $$$${of}\:\left({S}\right)\:{if}\:{and}\:{only}\:{a};{b};{c}\:{are}\: \\ $$$${roots}\:{of}\:{P}. \\ $$…