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AlgebraQuestion and Answers: Page 66

Question Number 194065 Answers: 0 Comments: 0

Question Number 194057 Answers: 0 Comments: 0

find all functions such that f(x)f(y) = x^a f((y/2))+y^b f((x/2))

$${find}\:{all}\:{functions}\:{such}\:{that} \\ $$$${f}\left({x}\right){f}\left({y}\right)\:=\:{x}^{{a}} {f}\left(\frac{{y}}{\mathrm{2}}\right)+{y}^{{b}} {f}\left(\frac{{x}}{\mathrm{2}}\right) \\ $$

Question Number 194037 Answers: 3 Comments: 0

Let a_1 ,a_2 ....a_n ∈R^+ , a_1 +a_2 +.....a_n =1 prove that: (a_1 /(2−a_1 ))+(a_2 /(2−a_2 )).......(a_n /(2−a_n ))≥(n/(2n−1))

$$ \\ $$$${Let}\:{a}_{\mathrm{1}} ,{a}_{\mathrm{2}} ....{a}_{{n}} \in{R}^{+} ,\:{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +.....{a}_{{n}} =\mathrm{1} \\ $$$${prove}\:{that}: \\ $$$$\frac{{a}_{\mathrm{1}} }{\mathrm{2}−{a}_{\mathrm{1}} }+\frac{{a}_{\mathrm{2}} }{\mathrm{2}−{a}_{\mathrm{2}} }.......\frac{{a}_{{n}} }{\mathrm{2}−{a}_{{n}} }\geqslant\frac{{n}}{\mathrm{2}{n}−\mathrm{1}} \\ $$

Question Number 193978 Answers: 1 Comments: 0

Question Number 193972 Answers: 0 Comments: 4

Question Number 193965 Answers: 2 Comments: 0

a,b,c are positive real numbers And (1/(a+b+1))+(1/(b+c+1))+(1/(a+c+1))≥1 prove that a+b+c≥ab+bc+ac

$${a},{b},{c}\:{are}\:{positive}\:{real}\:{numbers} \\ $$$${And} \\ $$$$\frac{\mathrm{1}}{{a}+{b}+\mathrm{1}}+\frac{\mathrm{1}}{{b}+{c}+\mathrm{1}}+\frac{\mathrm{1}}{{a}+{c}+\mathrm{1}}\geqslant\mathrm{1} \\ $$$${prove}\:{that}\:{a}+{b}+{c}\geqslant{ab}+{bc}+{ac} \\ $$

Question Number 193962 Answers: 2 Comments: 0

Question Number 194691 Answers: 0 Comments: 0

a, b, c≥0, a+b+c=2. Prove that 3a+8ab+16abc≤12.

$${a},\:{b},\:{c}\geqslant\mathrm{0},\:{a}+{b}+{c}=\mathrm{2}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{3}{a}+\mathrm{8}{ab}+\mathrm{16}{abc}\leqslant\mathrm{12}. \\ $$

Question Number 193922 Answers: 4 Comments: 0

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Question Number 193875 Answers: 2 Comments: 0

a,b,c,d,e,f, are + real numbers prove: (a/(b+c))+(b/(c+d))+(c/(d+e))+(d/(e+f))+(e/(f+a))+(f/(a+b))≥3

$${a},{b},{c},{d},{e},{f},\:{are}\:+\:{real}\:{numbers} \\ $$$${prove}: \\ $$$$\frac{{a}}{{b}+{c}}+\frac{{b}}{{c}+{d}}+\frac{{c}}{{d}+{e}}+\frac{{d}}{{e}+{f}}+\frac{{e}}{{f}+{a}}+\frac{{f}}{{a}+{b}}\geqslant\mathrm{3} \\ $$

Question Number 193848 Answers: 2 Comments: 0

Question Number 193796 Answers: 0 Comments: 0

Let G be a finite group,f be an automorphism of G such that f(x)=x ⇒x=e . Then prove that, (i)∀g∈G, ∃x∈G such that g=x^(−1) f(x). (ii)If ∀x∈G , f(f(x))=x ⇒ G is Abelian.

$${Let}\:{G}\:{be}\:{a}\:{finite}\:{group},{f}\:{be}\:{an}\:{automorphism}\:{of}\:{G} \\ $$$${such}\:{that}\:{f}\left({x}\right)={x}\:\Rightarrow{x}={e}\:. \\ $$$${Then}\:{prove}\:{that}, \\ $$$$\left(\boldsymbol{{i}}\right)\forall{g}\in{G},\:\exists{x}\in{G}\:{such}\:{that}\:{g}={x}^{−\mathrm{1}} {f}\left({x}\right). \\ $$$$\left(\boldsymbol{{ii}}\right){If}\:\forall{x}\in{G}\:,\:{f}\left({f}\left({x}\right)\right)={x}\:\Rightarrow\:{G}\:{is}\:{Abelian}. \\ $$$$ \\ $$

Question Number 193794 Answers: 1 Comments: 0

Let H be a subgroup of (R,+) such that H∩[−1,1] contains a non zero element. Prove that H is cyclic.

$${Let}\:{H}\:{be}\:{a}\:{subgroup}\:{of}\:\left(\mathbb{R},+\right)\:{such}\:{that}\:{H}\cap\left[−\mathrm{1},\mathrm{1}\right]\: \\ $$$${contains}\:{a}\:{non}\:{zero}\:{element}. \\ $$$${Prove}\:{that}\:{H}\:{is}\:{cyclic}. \\ $$

Question Number 193793 Answers: 1 Comments: 4

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Question Number 193714 Answers: 1 Comments: 3

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Question Number 193651 Answers: 0 Comments: 1

determiner rayon R

$$\mathrm{determiner}\:\mathrm{rayon}\:\boldsymbol{\mathrm{R}} \\ $$

Question Number 193647 Answers: 2 Comments: 0

Question Number 193565 Answers: 1 Comments: 0

Question Number 193546 Answers: 2 Comments: 0

if a+b+c=1 find maximum ab +bc +ca a , b , c are non negative integers

$${if}\:{a}+{b}+{c}=\mathrm{1} \\ $$$${find}\:{maximum}\:{ab}\:+{bc}\:+{ca}\: \\ $$$${a}\:,\:{b}\:,\:{c}\:{are}\:{non}\:{negative}\:{integers} \\ $$$$ \\ $$

Question Number 193543 Answers: 0 Comments: 4

solve

$${solve} \\ $$

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