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Question Number 155069 Answers: 0 Comments: 0

if x;y;z>0 such that x+y+z=3 and 0≤𝛌≤1 then prove that: (x/(y^2 +λ)) + (y/(z^2 +λ)) + (z/(x^2 +λ)) ≥ (3/(λ+1))

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{3} \\ $$$$\mathrm{and}\:\:\mathrm{0}\leqslant\boldsymbol{\lambda}\leqslant\mathrm{1}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}}{\mathrm{y}^{\mathrm{2}} +\lambda}\:+\:\frac{\mathrm{y}}{\mathrm{z}^{\mathrm{2}} +\lambda}\:+\:\frac{\mathrm{z}}{\mathrm{x}^{\mathrm{2}} +\lambda}\:\geqslant\:\frac{\mathrm{3}}{\lambda+\mathrm{1}} \\ $$

Question Number 155068 Answers: 0 Comments: 0

Question Number 155257 Answers: 1 Comments: 5

Question Number 155206 Answers: 0 Comments: 0

Determine all positive integers a;b;c;d;x;y;z;t and a≠b≠c≠d which satisfy a+b+c=td ; b+c+d=xa ; c+d+a=yb ; d+a+b=zc

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{a};\mathrm{b};\mathrm{c};\mathrm{d};\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}\:\:\mathrm{and}\:\:\mathrm{a}\neq\mathrm{b}\neq\mathrm{c}\neq\mathrm{d} \\ $$$$\mathrm{which}\:\mathrm{satisfy}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{td}\:; \\ $$$$\mathrm{b}+\mathrm{c}+\mathrm{d}=\mathrm{xa}\:;\:\mathrm{c}+\mathrm{d}+\mathrm{a}=\mathrm{yb}\:;\:\mathrm{d}+\mathrm{a}+\mathrm{b}=\mathrm{zc} \\ $$

Question Number 155207 Answers: 1 Comments: 0

𝛀 =∫_( 0) ^( ∞) ((ln(x))/(e^x + e^(−x) )) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{e}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{e}^{−\boldsymbol{\mathrm{x}}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 155058 Answers: 0 Comments: 0

Question Number 155056 Answers: 1 Comments: 0

Question Number 155039 Answers: 2 Comments: 0

(2/5) + (6/5) = ? Hihi

$$\frac{\mathrm{2}}{\mathrm{5}}\:+\:\frac{\mathrm{6}}{\mathrm{5}}\:=\:?\:\:{Hihi} \\ $$

Question Number 155036 Answers: 1 Comments: 0

If a^b =b^a and a=2b then find the value of a^2 +b^2 ?

$$\:\mathrm{If}\:\:\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{b}}} =\boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{a}}} \:\mathrm{and}\:\boldsymbol{\mathrm{a}}=\mathrm{2}\boldsymbol{\mathrm{b}}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\:\:\mathrm{value}\:\mathrm{of}\:\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} \:? \\ $$

Question Number 155035 Answers: 1 Comments: 0

Question Number 155034 Answers: 3 Comments: 4

Question Number 155033 Answers: 1 Comments: 0

Question Number 155032 Answers: 1 Comments: 0

Question Number 155023 Answers: 1 Comments: 2

Question Number 155016 Answers: 0 Comments: 0

∫_0 ^∞ tan^2 u(1−(1/((u+1)^2 )))du

$$\int_{\mathrm{0}} ^{\infty} {tan}^{\mathrm{2}} {u}\left(\mathrm{1}−\frac{\mathrm{1}}{\left({u}+\mathrm{1}\right)^{\mathrm{2}} }\right){du} \\ $$

Question Number 155013 Answers: 2 Comments: 0

∫_0 ^1 ln(−lnx)(x^(μ−1) /( (√(−ln(x)))))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(−{lnx}\right)\frac{{x}^{\mu−\mathrm{1}} }{\:\sqrt{−{ln}\left({x}\right)}}{dx}=? \\ $$

Question Number 155017 Answers: 1 Comments: 0

∫_0 ^(3/2) E(x^2 )dx; avec E(x) la partie entiere

$$\int_{\mathrm{0}} ^{\frac{\mathrm{3}}{\mathrm{2}}} {E}\left({x}^{\mathrm{2}} \right){dx};\:{avec}\:{E}\left({x}\right)\:{la}\:{partie}\:{entiere} \\ $$

Question Number 154994 Answers: 1 Comments: 0

Find ∫_( 0) ^( 1) x^2 tan^(-1) (2x)ln^2 (3x)dx=?

$$\mathrm{Find} \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{tan}^{-\mathrm{1}} \left(\mathrm{2x}\right)\mathrm{ln}^{\mathrm{2}} \left(\mathrm{3x}\right)\mathrm{dx}=? \\ $$

Question Number 154993 Answers: 0 Comments: 0

Question Number 154992 Answers: 1 Comments: 0

Find the general solution of cos 2θ +cos θ = sin θ

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\: \\ $$$$\mathrm{cos}\:\mathrm{2}\theta\:+\mathrm{cos}\:\theta\:=\:\mathrm{sin}\:\theta \\ $$

Question Number 154989 Answers: 2 Comments: 0

x_1 and x_2 is root log_2 x^((1+^2 log x)) =2, the value is x_1 +x_2 = ... a. 2(1/4) b. 2(1/2) c. 4(1/4) d. 4(1/2) e. 6(1/4)

$${x}_{\mathrm{1}} \:{and}\:{x}_{\mathrm{2}} \:{is}\:{root}\:\mathrm{log}_{\mathrm{2}} {x}^{\left(\mathrm{1}+^{\mathrm{2}} \mathrm{log}\:{x}\right)} =\mathrm{2},\:{the}\:{value} \\ $$$${is}\:{x}_{\mathrm{1}} +{x}_{\mathrm{2}} =\:... \\ $$$${a}.\:\mathrm{2}\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${b}.\:\mathrm{2}\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${c}.\:\mathrm{4}\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${d}.\:\mathrm{4}\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${e}.\:\mathrm{6}\frac{\mathrm{1}}{\mathrm{4}} \\ $$

Question Number 154987 Answers: 1 Comments: 1

∫_0 ^( ∞) (x^c /c^x ) dx

$$\int_{\mathrm{0}} ^{\:\infty} \:\frac{{x}^{{c}} }{{c}^{{x}} }\:\:{dx} \\ $$

Question Number 154986 Answers: 2 Comments: 0

Solve the system in R 2x = ((y^2 - 4y + 1)/(y^2 - y + 1)) y = ((-x^2 + 6x - 1)/(3x^2 - 2x + 3))

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{in}\:\mathbb{R} \\ $$$$\mathrm{2x}\:=\:\frac{\mathrm{y}^{\mathrm{2}} \:-\:\mathrm{4y}\:+\:\mathrm{1}}{\mathrm{y}^{\mathrm{2}} \:-\:\mathrm{y}\:+\:\mathrm{1}} \\ $$$$\mathrm{y}\:=\:\frac{-\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{6x}\:-\:\mathrm{1}}{\mathrm{3x}^{\mathrm{2}} \:-\:\mathrm{2x}\:+\:\mathrm{3}} \\ $$

Question Number 154980 Answers: 1 Comments: 0

if a and b are real numbers determine a necessary and sufficient condition so as the equation x^2 + ((a(√b))/x) = a + b to have three real distinct roots.

$$\mathrm{if}\:\:\boldsymbol{\mathrm{a}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{b}}\:\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{determine}\:\mathrm{a}\:\mathrm{necessary}\:\mathrm{and}\:\mathrm{sufficient} \\ $$$$\mathrm{condition}\:\mathrm{so}\:\mathrm{as}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\frac{\mathrm{a}\sqrt{\mathrm{b}}}{\mathrm{x}}\:=\:\mathrm{a}\:+\:\mathrm{b} \\ $$$$\mathrm{to}\:\mathrm{have}\:\mathrm{three}\:\mathrm{real}\:\mathrm{distinct}\:\mathrm{roots}. \\ $$

Question Number 154976 Answers: 0 Comments: 0

Question Number 154975 Answers: 0 Comments: 0

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