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Question Number 194808 Answers: 0 Comments: 4

suppose a,b,c are positive real numbers prove the inequality (((a+b)/2))(((b+c)/2))(((c+a)/2))≥(((a+b+c)/3))(((abc)^2 ))^(1/3)

$$ \\ $$$${suppose}\:{a},{b},{c}\:{are}\:{positive}\:{real}\:{numbers} \\ $$$${prove}\:{the}\:{inequality} \\ $$$$\left(\frac{{a}+{b}}{\mathrm{2}}\right)\left(\frac{{b}+{c}}{\mathrm{2}}\right)\left(\frac{{c}+{a}}{\mathrm{2}}\right)\geqslant\left(\frac{{a}+{b}+{c}}{\mathrm{3}}\right)\sqrt[{\mathrm{3}}]{\left({abc}\right)^{\mathrm{2}} } \\ $$

Question Number 194796 Answers: 1 Comments: 0

A 1m^2 rectangle which length is less than 1 is a square. Why?

$${A}\:\mathrm{1}{m}^{\mathrm{2}} \:{rectangle}\:{which}\:{length}\:{is}\: \\ $$$${less}\:{than}\:\mathrm{1}\:{is}\:\:{a}\:{square}.\:{Why}? \\ $$

Question Number 194791 Answers: 1 Comments: 0

of x (((x−(√2)))^(1/7) /2) −(((x−(√2)))^(1/7) /x^2 ) = (x/2) ((x^2 /(x+(√2))))^(1/7)

$$\:\: \: \: \:{of}\:{x}\: \\ $$$$\:\:\frac{\sqrt[{\mathrm{7}}]{{x}−\sqrt{\mathrm{2}}}}{\mathrm{2}}\:−\frac{\sqrt[{\mathrm{7}}]{{x}−\sqrt{\mathrm{2}}}}{{x}^{\mathrm{2}} }\:=\:\frac{{x}}{\mathrm{2}}\:\sqrt[{\mathrm{7}}]{\frac{{x}^{\mathrm{2}} }{{x}+\sqrt{\mathrm{2}}}}\:\:\: \\ $$

Question Number 194790 Answers: 1 Comments: 0

Question Number 194786 Answers: 1 Comments: 0

x^n +y^n =¿ (n∈N^∗ )

$${x}^{{n}} +{y}^{{n}} =¿\:\left({n}\in{N}^{\ast} \right) \\ $$

Question Number 194785 Answers: 0 Comments: 0

∫∫ x^2 +y^2 dxdy (D=x^4 +y^4 ≤1)

$$\int\int\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:{dxdy}\:\left({D}={x}^{\mathrm{4}} +{y}^{\mathrm{4}} \leqslant\mathrm{1}\right) \\ $$

Question Number 194781 Answers: 0 Comments: 0

f_(n ) the general sentence is seqiencee fibonacci. prove that : f_(2n−1) =f_n ^2 +f_(n−1) ^2

$${f}_{{n}\:} \:\:{the}\:{general}\:{sentence}\:{is}\:{seqiencee} \\ $$$${fibonacci}.\: \\ $$$${prove}\:{that}\::\:\:{f}_{\mathrm{2}{n}−\mathrm{1}} ={f}_{{n}} ^{\mathrm{2}} +{f}_{{n}−\mathrm{1}} ^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 194779 Answers: 1 Comments: 4

If a divided by b gives q remaining r Then (a/b) = q,rrr... in base b+1

$${If}\:\:{a}\:\:{divided}\:{by}\:{b}\:{gives}\:{q}\:\:{remaining}\:{r} \\ $$$${Then}\:\:\frac{{a}}{{b}}\:=\:{q},{rrr}...\:\:{in}\:{base}\:{b}+\mathrm{1} \\ $$

Question Number 194767 Answers: 2 Comments: 0

tan θ = 2 ((8sin θ+5cos θ)/(sin^3 θ+cos^3 θ+cos θ)) =?

$$\:\:\:\:\:\mathrm{tan}\:\theta\:=\:\mathrm{2}\: \\ $$$$\:\:\:\frac{\mathrm{8sin}\:\theta+\mathrm{5cos}\:\theta}{\mathrm{sin}\:^{\mathrm{3}} \theta+\mathrm{cos}\:^{\mathrm{3}} \theta+\mathrm{cos}\:\theta}\:=? \\ $$

Question Number 194766 Answers: 2 Comments: 0

1+2cot 2x cot x = 3 x=?

$$\:\:\:\mathrm{1}+\mathrm{2cot}\:\mathrm{2}{x}\:\mathrm{cot}\:{x}\:=\:\mathrm{3}\: \\ $$$$\:\:\:{x}=? \\ $$

Question Number 194759 Answers: 1 Comments: 1

∫_0 ^( 1) ∫_0 ^( 1) (((1+x^2 )/(1+x^2 +y^2 ))) dxdy

$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\left(\frac{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\right)\:\mathrm{dxdy}\: \\ $$

Question Number 194756 Answers: 3 Comments: 0

((x−a)/( (√x) +(√a))) = (((√x)−(√a))/3) +2(√a)

$$\:\:\:\:\: \\ $$$$ \:\frac{\mathrm{x}−\mathrm{a}}{\:\sqrt{\mathrm{x}}\:+\sqrt{\mathrm{a}}}\:=\:\frac{\sqrt{\mathrm{x}}−\sqrt{\mathrm{a}}}{\mathrm{3}}\:+\mathrm{2}\sqrt{\mathrm{a}}\: \\ $$

Question Number 194736 Answers: 0 Comments: 2

( log _(sin x cos x) (cos x))( log _(sin x cos x) (sin x))=(1/4)

$$\:\:\:\: \\ $$$$ \left(\:\mathrm{log}\:_{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}} \:\left(\mathrm{cos}\:\mathrm{x}\right)\right)\left(\:\mathrm{log}\:_{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}} \left(\mathrm{sin}\:\mathrm{x}\right)\right)=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\:\: \\ $$

Question Number 194735 Answers: 1 Comments: 0

Question Number 194732 Answers: 2 Comments: 1

Question Number 194715 Answers: 0 Comments: 0

Question Number 194713 Answers: 0 Comments: 2

Question Number 194709 Answers: 1 Comments: 0

Show that in fibonacci sequence f_(3n) =f_n ^3 +f_(n+1) ^3 −f_(n−1) ^3

$${Show}\:{that}\:\:{in}\:{fibonacci}\:{sequence} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{f}_{\mathrm{3}{n}} ={f}_{{n}} ^{\mathrm{3}} +{f}_{{n}+\mathrm{1}} ^{\mathrm{3}} −{f}_{{n}−\mathrm{1}} ^{\mathrm{3}} \\ $$$$ \\ $$

Question Number 194710 Answers: 0 Comments: 21

let p be a prime number & let a_1 ,a_2 ,a_3 ,...,a_(p ) be integers show that , there exists an integer k such that the numbers a_1 +k, a_2 +k,a_3 +k,....,a_p +k produce at least (1/2)p distinct remainders when divided by p.

$${let}\:{p}\:{be}\:{a}\:{prime}\:{number} \\ $$$$\&\:{let}\:{a}_{\mathrm{1}} \:,{a}_{\mathrm{2}} ,{a}_{\mathrm{3}} ,...,{a}_{{p}\:} {be}\:{integers} \\ $$$${show}\:{that}\:,\:{there}\:{exists}\:{an}\:{integer}\:{k}\:{such}\:{that}\:{the}\:{numbers} \\ $$$${a}_{\mathrm{1}} +{k},\:{a}_{\mathrm{2}} +{k},{a}_{\mathrm{3}} +{k},....,{a}_{{p}} +{k} \\ $$$${produce}\:{at}\:{least}\:\frac{\mathrm{1}}{\mathrm{2}}{p}\:{distinct}\:{remainders} \\ $$$${when}\:{divided}\:{by}\:{p}. \\ $$

Question Number 194700 Answers: 0 Comments: 2

Question Number 194697 Answers: 2 Comments: 0

((tan x)/(tan x−tan 3x)) = (1/3) then ((cot x)/(cot x+cot 3x)) =?

$$\:\:\: \frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{tan}\:\mathrm{x}−\mathrm{tan}\:\mathrm{3x}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{then} \\ $$$$\:\:\:\frac{\mathrm{cot}\:\mathrm{x}}{\mathrm{cot}\:\mathrm{x}+\mathrm{cot}\:\mathrm{3x}}\:=? \\ $$

Question Number 194695 Answers: 1 Comments: 0

(x/(a+b−c)) =(y/(b+c−a))=(z/(c+a−b)) Then (a−b)x+(b−c)y+(c−a)z =?

$$\:\:\:\: \:\frac{\mathrm{x}}{\mathrm{a}+\mathrm{b}−\mathrm{c}}\:=\frac{\mathrm{y}}{\mathrm{b}+\mathrm{c}−\mathrm{a}}=\frac{\mathrm{z}}{\mathrm{c}+\mathrm{a}−\mathrm{b}} \\ $$$$\:\mathrm{Then}\:\left(\mathrm{a}−\mathrm{b}\right)\mathrm{x}+\left(\mathrm{b}−\mathrm{c}\right)\mathrm{y}+\left(\mathrm{c}−\mathrm{a}\right)\mathrm{z}\:=? \\ $$

Question Number 194693 Answers: 1 Comments: 0

if f_n =f_(n−1) +f_(n−2) ; f_1 =f_2 =1 then prove that 5∣f_(5n)

$${if}\:\:\:{f}_{{n}} ={f}_{{n}−\mathrm{1}} +{f}_{{n}−\mathrm{2}} \:\:;\:\:{f}_{\mathrm{1}} ={f}_{\mathrm{2}} =\mathrm{1} \\ $$$${then}\:\:\:{prove}\:{that}\:\:\:\mathrm{5}\mid{f}_{\mathrm{5}{n}} \:\: \\ $$

Question Number 194685 Answers: 1 Comments: 0

Question Number 194662 Answers: 0 Comments: 2

∫_0 ^(Π/2) (√(4sin^2 t+cos^2 t)) dt

$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}+{cos}^{\mathrm{2}} {t}}\:\:{dt} \\ $$

Question Number 194654 Answers: 1 Comments: 0

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