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AlgebraQuestion and Answers: Page 69

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 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 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 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 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 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 194648 Answers: 3 Comments: 3

Question Number 194637 Answers: 4 Comments: 1

x+y=1 x^2 +y^2 =2 x^(11) +y^(11) =?

$$ \\ $$$${x}+{y}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}^{\mathrm{11}} +{y}^{\mathrm{11}} =? \\ $$$$ \\ $$$$ \\ $$

Question Number 194636 Answers: 0 Comments: 3

Question Number 194634 Answers: 1 Comments: 0

a_1 ,a_2 ,a_3 ,....,a_n >0 such that a_i ∈[0,i] ∀ i∈{1,2,3,4,...,n} prove that 2^n .a_1 (a_1 +a_2 )...(a_1 +a_2 +...+a_n )≥(n+1)(a_1 ^2 .a_2 ^2 ...a_n ^2 )

$${a}_{\mathrm{1}} ,{a}_{\mathrm{2}} ,{a}_{\mathrm{3}} ,....,{a}_{{n}} >\mathrm{0}\:{such}\:{that}\:{a}_{{i}} \in\left[\mathrm{0},{i}\right]\: \\ $$$$\forall\:{i}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},...,{n}\right\}\:{prove}\:{that} \\ $$$$\mathrm{2}^{{n}} .{a}_{\mathrm{1}} \left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} \right)...\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{{n}} \right)\geqslant\left({n}+\mathrm{1}\right)\left({a}_{\mathrm{1}} ^{\mathrm{2}} .{a}_{\mathrm{2}} ^{\mathrm{2}} ...{a}_{{n}} ^{\mathrm{2}} \right) \\ $$

Question Number 194619 Answers: 1 Comments: 0

Find the sum of the roots of the equation: −3x^3 + 8x^2 − 6x − 7 = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}: \\ $$$$−\mathrm{3x}^{\mathrm{3}} \:+\:\mathrm{8x}^{\mathrm{2}} \:−\:\mathrm{6x}\:−\:\mathrm{7}\:=\:\mathrm{0} \\ $$

Question Number 194612 Answers: 1 Comments: 2

Question Number 194610 Answers: 1 Comments: 0

where can I learn about multiple sigma notaions of dependent and independent variables something like this Σ_(1≤i) Σ_(<j) Σ_(<k≤1) (i+j+k)=λ find λ I want to know what to study

$${where}\:{can}\:{I}\:{learn}\:{about}\:{multiple}\:{sigma}\:{notaions} \\ $$$${of}\:{dependent}\:{and}\:{independent}\:{variables} \\ $$$$ \\ $$$${something}\:{like}\:{this} \\ $$$$\underset{\mathrm{1}\leqslant{i}} {\sum}\underset{<{j}} {\sum}\underset{<{k}\leqslant\mathrm{1}} {\sum}\left({i}+{j}+{k}\right)=\lambda \\ $$$${find}\:\lambda \\ $$$${I}\:{want}\:{to}\:{know}\:{what}\:{to}\:{study} \\ $$

Question Number 194586 Answers: 1 Comments: 2

abc = e^3 + d^3 + f^3 edf = a^3 + b^3 + c^3 find: abc and edf

$$\mathrm{abc}\:=\:\mathrm{e}^{\mathrm{3}} \:+\:\mathrm{d}^{\mathrm{3}} \:+\:\mathrm{f}^{\mathrm{3}} \\ $$$$\mathrm{edf}\:=\:\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{c}^{\mathrm{3}} \\ $$$$\mathrm{find}:\:\mathrm{abc}\:\:\mathrm{and}\:\:\mathrm{edf}\: \\ $$

Question Number 194579 Answers: 2 Comments: 0

if u_n =(1/( (√5)))[(((1+(√5))/2))^n −(((1−(√5))/2))^n ] then u_(n+1) =u_n +u_(n−1) ? ; n=0,1,2,..

$${if}\:\:\:\:{u}_{{n}} =\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\left[\left(\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}} −\left(\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}} \right] \\ $$$$\:{then}\:\:\:{u}_{{n}+\mathrm{1}} ={u}_{{n}} +{u}_{{n}−\mathrm{1}} \:\:\:?\:\:\:\:\:;\:\:\:{n}=\mathrm{0},\mathrm{1},\mathrm{2},.. \\ $$

Question Number 194573 Answers: 0 Comments: 0