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Question Number 151193 Answers: 0 Comments: 0

Question Number 151192 Answers: 0 Comments: 1

Question Number 151189 Answers: 0 Comments: 0

In △ABC the following relationship holds: (𝛟-golden ratio) sinA + ((sinB)/𝛟) + ((sinC)/𝛟) < (1/𝛟) + ((1+(√𝛟)+𝛟)/(2𝛟))

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{the}\:\mathrm{following}\:\mathrm{relationship} \\ $$$$\mathrm{holds}:\:\left(\boldsymbol{\varphi}-\mathrm{golden}\:\mathrm{ratio}\right) \\ $$$$\mathrm{sinA}\:+\:\frac{\mathrm{sinB}}{\boldsymbol{\varphi}}\:+\:\frac{\mathrm{sinC}}{\boldsymbol{\varphi}}\:<\:\frac{\mathrm{1}}{\boldsymbol{\varphi}}\:+\:\frac{\mathrm{1}+\sqrt{\boldsymbol{\varphi}}+\boldsymbol{\varphi}}{\mathrm{2}\boldsymbol{\varphi}} \\ $$

Question Number 151182 Answers: 0 Comments: 0

∫_0 ^( ∞) ((sin(2x)ln(x))/x) dx= m.∫_0 ^( ∞) (( ln(1+2x+x^2 ))/(x(ln^2 (x)+ π^( 2) ))) dx m=?....

$$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\mathrm{2}{x}\right){ln}\left({x}\right)}{{x}}\:{dx}=\:{m}.\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}\left(\mathrm{1}+\mathrm{2}{x}+{x}^{\mathrm{2}} \right)}{{x}\left({ln}^{\mathrm{2}} \left({x}\right)+\:\pi^{\:\mathrm{2}} \right)}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:{m}=?.... \\ $$

Question Number 151181 Answers: 2 Comments: 0

if ∣x∣<1 find x−4x^2 +9x^3 −16x^4 +...

$$\mathrm{if}\:\:\mid\boldsymbol{\mathrm{x}}\mid<\mathrm{1} \\ $$$$\mathrm{find}\:\:\mathrm{x}−\mathrm{4x}^{\mathrm{2}} +\mathrm{9x}^{\mathrm{3}} −\mathrm{16x}^{\mathrm{4}} +... \\ $$

Question Number 151179 Answers: 2 Comments: 0

if ∣x∣<1 find x+2x^2 +3x^3 +...

$$\mathrm{if}\:\:\mid\boldsymbol{\mathrm{x}}\mid<\mathrm{1} \\ $$$$\mathrm{find}\:\:\mathrm{x}+\mathrm{2x}^{\mathrm{2}} +\mathrm{3x}^{\mathrm{3}} +... \\ $$

Question Number 151175 Answers: 0 Comments: 0

Question Number 151174 Answers: 2 Comments: 0

((a+(√(3∙((a+(√(3∙((a+(√(3∙...))))^(1/3) ))))^(1/3) ))))^(1/3) = 3 find a=?

$$\sqrt[{\mathrm{3}}]{{a}+\sqrt{\mathrm{3}\centerdot\sqrt[{\mathrm{3}}]{{a}+\sqrt{\mathrm{3}\centerdot\sqrt[{\mathrm{3}}]{{a}+\sqrt{\mathrm{3}\centerdot...}}}}}}\:\:=\:\mathrm{3}\: \\ $$$$\mathrm{find}\:\:{a}=? \\ $$

Question Number 151191 Answers: 0 Comments: 0

Question Number 151197 Answers: 1 Comments: 0

etude de la monotonie? svp u_n =Σ_(k=1) ^n (1/k)−ln(n)

$${etude}\:{de}\:{la}\:{monotonie}?\:{svp} \\ $$$${u}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}−{ln}\left({n}\right) \\ $$

Question Number 151196 Answers: 0 Comments: 0

Question Number 151164 Answers: 1 Comments: 1

Question Number 151162 Answers: 1 Comments: 2

Question Number 151157 Answers: 0 Comments: 0

$${x} \\ $$

Question Number 151142 Answers: 0 Comments: 0

if x;y;z∈R^+ and (1/x^2 ) + (1/y^2 ) + (1/z^2 ) = ((27)/4) prove that: ((x^3 + y^2 )/(x^2 + y^2 )) + ((y^3 + z^2 )/(y^2 + z^2 )) + ((z^3 + x^2 )/(z^2 + x^2 )) ≥ (5/2)

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}\in\mathbb{R}^{+} \:\:\mathrm{and}\:\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{y}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{2}} }\:=\:\frac{\mathrm{27}}{\mathrm{4}} \\ $$$$\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} }\:+\:\frac{\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{2}} }{\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} }\:+\:\frac{\mathrm{z}^{\mathrm{3}} \:+\:\mathrm{x}^{\mathrm{2}} }{\mathrm{z}^{\mathrm{2}} \:+\:\mathrm{x}^{\mathrm{2}} }\:\geqslant\:\frac{\mathrm{5}}{\mathrm{2}} \\ $$

Question Number 151132 Answers: 0 Comments: 1

Find the minimum value of x , if α<82^o .

$$ \\ $$$${Find}\:{the}\:{minimum}\:{value}\:{of}\:\:{x}\:,\:{if}\:\:\:\alpha<\mathrm{82}^{{o}} \:. \\ $$

Question Number 151118 Answers: 1 Comments: 0

∫_0 ^( ∞) (( arctan((x/2))+arctan(2x))/(x^( 2) +1))=^? (π^( 2) /4)

$$\:\: \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{arctan}\left(\frac{{x}}{\mathrm{2}}\right)+{arctan}\left(\mathrm{2}{x}\right)}{{x}^{\:\mathrm{2}} +\mathrm{1}}\overset{?} {=}\frac{\pi^{\:\mathrm{2}} }{\mathrm{4}} \\ $$$$ \\ $$

Question Number 151117 Answers: 0 Comments: 0

Question Number 151115 Answers: 3 Comments: 0

Solve the system: { ((x^2 y^2 + xy^2 + x + y = 0)),((x^2 y + xy + 1 = 0)) :}

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}: \\ $$$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} \:+\:\mathrm{xy}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{y}\:=\:\mathrm{0}}\\{\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\:\mathrm{xy}\:+\:\mathrm{1}\:=\:\mathrm{0}}\end{cases} \\ $$

Question Number 151114 Answers: 1 Comments: 0

solve.... Q := ∫_0 ^( 1) ln (x ). ln ( 2− x )dx =? ...........■ m.n.1970...

$$ \\ $$$$\:\:\:\:\:{solve}.... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Q}\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\:\left({x}\:\right).\:{ln}\:\left(\:\mathrm{2}−\:{x}\:\right){dx}\:=?\:...........\blacksquare \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970}... \\ $$$$ \\ $$

Question Number 151111 Answers: 1 Comments: 0

Question Number 151104 Answers: 1 Comments: 1

Question Number 151101 Answers: 1 Comments: 0

Solve for real numbers the equation [(x/2)] + [((3x)/5)] = [(x/(10))] + x , where we denoting by [x] the great integer part of x.

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\left[\frac{\mathrm{x}}{\mathrm{2}}\right]\:+\:\left[\frac{\mathrm{3x}}{\mathrm{5}}\right]\:=\:\left[\frac{\mathrm{x}}{\mathrm{10}}\right]\:+\:\mathrm{x}\:,\:\:\mathrm{where}\:\mathrm{we} \\ $$$$\mathrm{denoting}\:\mathrm{by}\:\left[\boldsymbol{\mathrm{x}}\right]\:\mathrm{the}\:\mathrm{great}\:\mathrm{integer}\:\mathrm{part} \\ $$$$\mathrm{of}\:\boldsymbol{\mathrm{x}}. \\ $$

Question Number 151100 Answers: 1 Comments: 0

if 0≤x;y;z≤k and k>0 then: y(x - z) - z(x - k) ≤ k^2

$$\mathrm{if}\:\:\mathrm{0}\leqslant\mathrm{x};\mathrm{y};\mathrm{z}\leqslant\mathrm{k}\:\:\mathrm{and}\:\:\mathrm{k}>\mathrm{0}\:\:\mathrm{then}: \\ $$$$\mathrm{y}\left(\mathrm{x}\:-\:\mathrm{z}\right)\:-\:\mathrm{z}\left(\mathrm{x}\:-\:\mathrm{k}\right)\:\leqslant\:\mathrm{k}^{\mathrm{2}} \\ $$

Question Number 151097 Answers: 2 Comments: 0

Find sum of this expression . ((n),(1) ) + 2 ((n),(2) ) + 3 ((n),(3) ) + 4 ((n),(4) ) + … + n ((n),(n) ) Please show your workings. Thank you .

$${Find}\:\:{sum}\:\:{of}\:\:{this}\:\:{expression}\:. \\ $$$$\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\:+\:\mathrm{2}\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}\:+\:\mathrm{3}\begin{pmatrix}{{n}}\\{\mathrm{3}}\end{pmatrix}\:+\:\mathrm{4}\begin{pmatrix}{{n}}\\{\mathrm{4}}\end{pmatrix}\:+\:\ldots\:+\:{n}\begin{pmatrix}{{n}}\\{{n}}\end{pmatrix} \\ $$$${Please}\:\:{show}\:\:{your}\:\:{workings}.\:{Thank}\:\:{you}\:. \\ $$

Question Number 151096 Answers: 0 Comments: 4

find the center of gravity with respect to point O

$$ \\ $$find the center of gravity with respect to point O

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