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Question Number 171563 Answers: 1 Comments: 0
$$\:\:\:\:\:\:\:\:{evaluate}\:\:\: \\ $$$$\:\:\:\:\:\frac{\sqrt{\frac{\left({a}−{b}\right)^{\mathrm{7}} \:+\:\left({b}−{c}\right)^{\mathrm{7}} \:+\:\left({c}−{a}\right)^{\mathrm{7}} }{\left({a}−{b}\right)^{\mathrm{3}} \:+\:\left({b}−{c}\right)^{\mathrm{3}} \:+\:\left({c}−{a}\right)^{\mathrm{3}} }}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}}\:=\:\:?? \\ $$
Question Number 171558 Answers: 0 Comments: 0
$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:,\:\mathrm{I}-\mathrm{incenter} \\ $$$$\mathrm{ID}\bot\mathrm{BC}\:,\:\mathrm{IE}\bot\mathrm{CA}\:,\:\mathrm{IF}\bot\mathrm{AB} \\ $$$$\mathrm{D}\in\left(\mathrm{BC}\right)\:,\:\mathrm{E}\in\left(\mathrm{CA}\right)\:,\:\mathrm{F}\in\left(\mathrm{AB}\right) \\ $$$$\mathrm{I}_{\boldsymbol{\mathrm{a}}} \:,\:\mathrm{I}_{\boldsymbol{\mathrm{b}}} \:,\:\mathrm{I}_{\boldsymbol{\mathrm{c}}} -\mathrm{excenters}.\:\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{EF}}{\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}}\:\:+\:\:\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\frac{\mathrm{EF}}{\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}}\:\:=\:\:\frac{\mathrm{1}\:+\:\mathrm{4}\boldsymbol{\mathrm{r}}^{\mathrm{2}} }{\mathrm{R}}\:\centerdot\:\left[\mathrm{I}_{\boldsymbol{\mathrm{a}}} \mathrm{I}_{\boldsymbol{\mathrm{b}}} \mathrm{I}_{\boldsymbol{\mathrm{c}}} \right] \\ $$
Question Number 171552 Answers: 2 Comments: 0
Question Number 171593 Answers: 0 Comments: 2
$$\:{The}\:{maximum}\:{value}\:{of}\:{the} \\ $$$${expression}\:\mid\sqrt{\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{2}{a}^{\mathrm{2}} }\:−\sqrt{\mathrm{2}{a}^{\mathrm{2}} −\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}}\:\mid\: \\ $$$${where}\:{a}\:{and}\:{x}\:{real}\:{numbers}\:{is}−−− \\ $$
Question Number 171549 Answers: 1 Comments: 0
$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\begin{cases}{\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{3y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{7}}\\{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\sqrt{\mathrm{2}}\:\mathrm{z}\:\left(\mathrm{x}\:+\:\mathrm{y}\right)}\end{cases} \\ $$
Question Number 171546 Answers: 0 Comments: 1
$${f}\left({x}\right)=\frac{−{ln}\mid{x}\mid}{{x}}+{x}−\mathrm{2}\:\:,\:\:\:{g}\left({x}\right)=−{x}^{\mathrm{2}} +\mathrm{1}−{ln}\mid{x}\mid \\ $$$$ \\ $$Calculate the derivative of f(x) as a function of g(x)
Question Number 171530 Answers: 0 Comments: 0
Question Number 171529 Answers: 4 Comments: 1
Question Number 171528 Answers: 1 Comments: 0
Question Number 171527 Answers: 0 Comments: 0
Question Number 171519 Answers: 0 Comments: 0
Question Number 171518 Answers: 0 Comments: 2
Question Number 176847 Answers: 3 Comments: 0
$$\mathrm{If}\:{x},\:{y}\:\in\mathbb{Z} \\ $$$$\mathrm{23}!=\mathrm{2}^{{x}} \mathrm{5}^{{y}} {k} \\ $$$$\mathrm{and}\:{k}\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{number},\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\: \\ $$$$\mathrm{max}\left({x}+{y}\right)\:? \\ $$
Question Number 176846 Answers: 1 Comments: 0
$${m}\in\mathbb{Z} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{leading}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\mathrm{P}\left({x}\right)=\mathrm{4}{x}^{\frac{\mathrm{13}}{{m}−\mathrm{5}}} −\mathrm{6}{x}^{\mathrm{25}−\mathrm{2}{m}} +\mathrm{4}{x}^{\mathrm{13}} +\mathrm{5}{x}^{\mathrm{10}} −\mathrm{4}\:? \\ $$
Question Number 171514 Answers: 0 Comments: 0
$$\:{compute}\:{by}\:{betta}\:{function}\:\int_{\mathrm{0}} ^{\:\mathrm{4}\pi} \sqrt{{cscx}}\:{dx} \\ $$
Question Number 171512 Answers: 1 Comments: 0
Question Number 171507 Answers: 0 Comments: 0
Question Number 171502 Answers: 0 Comments: 1
$${Solve} \\ $$$$\frac{{dy}}{{dx}}\left({x}\mathrm{cos}\:{y}+{a}\mathrm{sin}\:\mathrm{2}{y}\right)=−\mathrm{1} \\ $$$$\frac{{dy}}{{dx}}=\left(\frac{{y}+\mathrm{2}}{{x}+{y}−\mathrm{1}}\right)^{\mathrm{2}} \\ $$
Question Number 171499 Answers: 1 Comments: 0
$$\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{sample}}\:\boldsymbol{\mathrm{of}}\:\mathrm{20}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{is}} \\ $$$$\mathrm{320}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{squares}}\: \\ $$$$\:\boldsymbol{\mathrm{is}}\:\mathrm{5840}.\:\boldsymbol{\mathrm{Calculate}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{mean}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\: \\ $$$$\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{nineteen}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{the}}\:\mathrm{20}^{\boldsymbol{\mathrm{th}}} \: \\ $$$$\boldsymbol{\mathrm{observation}}\:\boldsymbol{\mathrm{is}}\:\mathrm{25}. \\ $$
Question Number 171497 Answers: 0 Comments: 3
Question Number 171493 Answers: 1 Comments: 0
Question Number 171491 Answers: 0 Comments: 0
Question Number 171490 Answers: 0 Comments: 0
Question Number 171479 Answers: 1 Comments: 5
$$\:\boldsymbol{{tan}}^{\mathrm{2}} \frac{\boldsymbol{\pi}}{\mathrm{7}}\:+\boldsymbol{{tan}}^{\mathrm{2}} \frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{7}}\:+\boldsymbol{{tan}}^{\mathrm{2}} \frac{\mathrm{5}\boldsymbol{\pi}}{\mathrm{7}}=? \\ $$
Question Number 171477 Answers: 0 Comments: 0
Question Number 171475 Answers: 1 Comments: 0
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