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Question Number 211605    Answers: 1   Comments: 1

Let a_1 ,a_2 ,…a_n Is n real numbers. All fall in the interval (−1,1) ________________________ (1)Prove that: Π_(1≤i,j≤n) ((1+a_i a_j )/(1−a_i a_j ))≥1 (2) Determine the necessary andsufficient conditions forl equaity in the inequality.

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Let}\:\boldsymbol{{a}}_{\mathrm{1}} ,\boldsymbol{{a}}_{\mathrm{2}} ,\ldots\boldsymbol{{a}}_{\boldsymbol{{n}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Is}\:\mathrm{n}\:\mathrm{real}\:\mathrm{numbers}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{All}\:\mathrm{fall}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval}\:\left(−\mathrm{1},\mathrm{1}\right) \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\left(\mathrm{1}\right)\mathrm{Prove}\:\mathrm{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{1}\leq\boldsymbol{{i}},\boldsymbol{{j}}\leq\boldsymbol{{n}}} {\prod}\frac{\mathrm{1}+\boldsymbol{{a}}_{\boldsymbol{{i}}} \boldsymbol{{a}}_{\boldsymbol{{j}}} }{\mathrm{1}−\boldsymbol{{a}}_{\boldsymbol{{i}}} \boldsymbol{{a}}_{\boldsymbol{{j}}} }\geq\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Determine}\:\mathrm{the}\:\mathrm{necessary}\: \\ $$$$\mathrm{andsufficient}\:\mathrm{conditions}\:\mathrm{forl} \\ $$$$\mathrm{equaity}\:\mathrm{in}\:\mathrm{the}\:\mathrm{inequality}. \\ $$$$ \\ $$

Question Number 211603    Answers: 3   Comments: 0

1.Given a regular tetrahedron ABCD with vertices A(0,0,0)B(a,0,0), C(0,a,0),and D(0,0,a).Calculate the volume V and the surface area S of this tetrahedron.

$$\mathrm{1}.\mathrm{Given}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{tetrahedron}\:\boldsymbol{{ABCD}} \\ $$$$\mathrm{with}\:\mathrm{vertices}\:\boldsymbol{{A}}\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\boldsymbol{{B}}\left(\boldsymbol{{a}},\mathrm{0},\mathrm{0}\right), \\ $$$$\boldsymbol{{C}}\left(\mathrm{0},\boldsymbol{{a}},\mathrm{0}\right),\boldsymbol{\mathrm{and}}\:\boldsymbol{{D}}\left(\mathrm{0},\mathrm{0},\boldsymbol{{a}}\right).\mathrm{Calculate}\:\mathrm{the} \\ $$$$\:\mathrm{volume}\:\boldsymbol{{V}}\:\:\mathrm{and}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{area}\:\boldsymbol{{S}}\:\boldsymbol{\mathrm{of}} \\ $$$$\mathrm{this}\:\mathrm{tetrahedron}. \\ $$

Question Number 211601    Answers: 0   Comments: 0

set 𝛂(2)^(1/3) ,ask Q(𝛂)Upper irreducible cubic equation: x^3 −3x+1=0 All the roots of

$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{set}\:\boldsymbol{\alpha}\sqrt[{\mathrm{3}}]{\mathrm{2}},\mathrm{ask}\:\mathbb{Q}\left(\boldsymbol{\alpha}\right)\mathrm{Upper}\:\mathrm{irreducible}\:\mathrm{cubic}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{x}}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{All}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of} \\ $$$$ \\ $$

Question Number 211598    Answers: 2   Comments: 0

{ ((x^2 −4x+3<0)),((((2x−1)/(x+2))≥1)) :}

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{{x}}+\mathrm{3}<\mathrm{0}}\\{\frac{\mathrm{2}\boldsymbol{{x}}−\mathrm{1}}{\boldsymbol{{x}}+\mathrm{2}}\geq\mathrm{1}}\end{cases} \\ $$$$ \\ $$

Question Number 211597    Answers: 1   Comments: 0

2x^4 −4x^3 −22x^2 +24x+36=0

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\boldsymbol{{x}}^{\mathrm{4}} −\mathrm{4}\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{22}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{24}\boldsymbol{{x}}+\mathrm{36}=\mathrm{0} \\ $$$$ \\ $$

Question Number 211595    Answers: 1   Comments: 0

Question Number 211594    Answers: 0   Comments: 1

If , H_n ^( −) =1−(1/2) +(1/3) −...+(((−1)^(n+1) )/n) prove that:Σ_(n=1) ^∞ ((H_n ^( − ) −ln(2))/n)=ln^2 (2) −−−−−−−−−−

$$ \\ $$$$\:{If}\:,\:\:\overset{\:\:−} {{H}}_{{n}} \:=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{3}}\:−...+\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{{n}}\:\:\: \\ $$$${prove}\:{that}:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\overset{\:\:−\:} {{H}}_{{n}} −\mathrm{ln}\left(\mathrm{2}\right)}{{n}}=\mathrm{ln}^{\mathrm{2}} \left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:−−−−−−−−−−\:\:\:\:\:\: \\ $$

Question Number 211579    Answers: 2   Comments: 0

∫(1/((1−x^4 )(√(1+x^2 ))))dx.

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{1}}{\left(\mathrm{1}−\boldsymbol{{x}}^{\mathrm{4}} \right)\sqrt{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} }}\boldsymbol{{dx}}. \\ $$$$ \\ $$

Question Number 211578    Answers: 2   Comments: 0

∫_0 ^(+∞) (x/( (√(1+x^4 ))))dx.

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{+\infty} \frac{\boldsymbol{{x}}}{\:\sqrt{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{4}} }}\boldsymbol{{dx}}. \\ $$$$ \\ $$

Question Number 211575    Answers: 2   Comments: 0

Question Number 211570    Answers: 0   Comments: 2

{x_n }>0 lim_(x→0) x_n ^(1/n) =a prove when a<1 lim_(x→0) x_n =0 when a>1 lim_(x→0) x_n =∞ and when a=1 what happen about x_n

$$\left\{{x}_{{n}} \right\}>\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} ={a} \\ $$$${prove}\:{when}\:{a}<\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} =\mathrm{0} \\ $$$${when}\:{a}>\mathrm{1}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} =\infty \\ $$$${and}\:{when}\:{a}=\mathrm{1}\:{what}\:{happen}\:{about}\:{x}_{{n}} \\ $$

Question Number 211567    Answers: 0   Comments: 2

if lim_(x→0) f(x)=lim_(x→0) g(x)=0 when do not use f(x) to replace g(x)

$${if}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{g}\left({x}\right)=\mathrm{0} \\ $$$${when}\:{do}\:{not}\:{use}\:{f}\left({x}\right)\:{to}\:\:{replace}\:{g}\left({x}\right)\:\:\: \\ $$

Question Number 211566    Answers: 0   Comments: 0

certificate: Σ_(n=−∞) ^(+∞) (1/(n^4 +a^4 ))=(𝛑/( (√2)a^3 )) ((sinh((√2)a𝛑)+sin((√2)a𝛑))/( (√2)a^3 cosh((√2)a𝛑)−cos((√2)a𝛑)))

$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\underset{\boldsymbol{{n}}=−\infty} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{4}} +\boldsymbol{{a}}^{\mathrm{4}} }=\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{2}}\boldsymbol{{a}}^{\mathrm{3}} }\:\frac{\boldsymbol{\mathrm{si}{n}\mathrm{h}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)+\boldsymbol{\mathrm{sin}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)}{\:\sqrt{\mathrm{2}}\boldsymbol{{a}}^{\mathrm{3}} \boldsymbol{\mathrm{cos}{h}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)−\boldsymbol{\mathrm{cos}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)} \\ $$$$ \\ $$

Question Number 211564    Answers: 0   Comments: 1

certificate:Σ_(x=1) ^(90) ((sin(x^2 ))/(sin(x)))=45

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}:\underset{\boldsymbol{{x}}=\mathrm{1}} {\overset{\mathrm{90}} {\sum}}\frac{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{{x}}^{\mathrm{2}} \right)}{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{{x}}\right)}=\mathrm{45} \\ $$$$ \\ $$$$ \\ $$

Question Number 211563    Answers: 1   Comments: 0

1.x^3 −3x−2=0 2.e^x +x^2 −4=0

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}.\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{x}}−\mathrm{2}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}.\boldsymbol{{e}}^{\boldsymbol{{x}}} +\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{4}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 211574    Answers: 1   Comments: 0

{ ((x^2 +y^2 =25)),((x+2y−3=0)) :}

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{25}}\\{\boldsymbol{{x}}+\mathrm{2}\boldsymbol{\mathrm{y}}−\mathrm{3}=\mathrm{0}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$

Question Number 211562    Answers: 2   Comments: 0

{ ((x^2 +y^2 =1)),((x^3 −y=0)) :}

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{1}}\\{\boldsymbol{{x}}^{\mathrm{3}} −\boldsymbol{\mathrm{y}}=\mathrm{0}}\end{cases} \\ $$

Question Number 211560    Answers: 0   Comments: 0

certificate: I=∫_0 ^(𝛑/2) (√(√(x^2 +ln^2 cos(x)−lncos(x)dx)))=(𝛑/2)(√(2ln2))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{I}}=\int_{\mathrm{0}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \sqrt{\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \boldsymbol{\mathrm{cos}}\left(\boldsymbol{{x}}\right)−\boldsymbol{\mathrm{lncos}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}}}=\frac{\boldsymbol{\pi}}{\mathrm{2}}\sqrt{\mathrm{2}\boldsymbol{\mathrm{ln}}\mathrm{2}} \\ $$$$ \\ $$

Question Number 211548    Answers: 2   Comments: 0

{ ((2x^2 +3y^2 −6xy=12)),((x^2 −y^2 =4)) :}

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\mathrm{y}}^{\mathrm{2}} −\mathrm{6}\boldsymbol{{x}\mathrm{y}}=\mathrm{12}}\\{\boldsymbol{{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{4}}\end{cases} \\ $$$$ \\ $$

Question Number 211546    Answers: 2   Comments: 0

∫(dx/( (√(x^2 −4x+13))))=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{\boldsymbol{{dx}}}{\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{{x}}+\mathrm{13}}}=? \\ $$

Question Number 211537    Answers: 3   Comments: 0

Question Number 211535    Answers: 1   Comments: 0

Question Number 211558    Answers: 1   Comments: 0

−−−−−−−−−−−− 𝛀= Σ_(n=0) ^∞ ((1/(3n+2)) −(1/(3n+1)) )= a𝛑 ⇒ a^2 = ? −−−−−−−−−−−−

$$ \\ $$$$ \\ $$$$\:\:\:\:\:−−−−−−−−−−−− \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\Omega}=\:\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\left(\frac{\mathrm{1}}{\mathrm{3}\boldsymbol{{n}}+\mathrm{2}}\:−\frac{\mathrm{1}}{\mathrm{3}\boldsymbol{{n}}+\mathrm{1}}\:\right)=\:\boldsymbol{{a}\pi} \\ $$$$\:\:\:\:\:\:\:\Rightarrow\:\:\boldsymbol{{a}}^{\mathrm{2}} =\:? \\ $$$$\:\:\:\:\:\:\:−−−−−−−−−−−− \\ $$

Question Number 211531    Answers: 1   Comments: 0

∫_0 ^∞ (x^2 /(sinh(x)^2 ))dx.

$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{{x}}^{\mathrm{2}} }{\boldsymbol{\mathrm{sinh}}\left(\boldsymbol{{x}}\right)^{\mathrm{2}} }\boldsymbol{{dx}}. \\ $$

Question Number 211529    Answers: 1   Comments: 0

known:x+y=3,ask: min((√(x^2 +1))+(√(y2−4)))=?

$$\mathrm{known}:\boldsymbol{{x}}+\boldsymbol{\mathrm{y}}=\mathrm{3},\mathrm{ask}: \\ $$$$\boldsymbol{\mathrm{min}}\left(\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}+\sqrt{\boldsymbol{\mathrm{y}}\mathrm{2}−\mathrm{4}}\right)=? \\ $$

Question Number 211528    Answers: 0   Comments: 2

I = ∫_0 ^( 1) (( tanh^(−1) (x^2 ))/x^( 2) ) dx= ?

$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{−\mathrm{1}} \:\left({x}^{\mathrm{2}} \:\right)}{{x}^{\:\mathrm{2}} }\:{dx}=\:?\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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