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AlgebraQuestion and Answers: Page 66
Question Number 194204 Answers: 1 Comments: 0
Question Number 194197 Answers: 1 Comments: 0
Question Number 194191 Answers: 1 Comments: 0
Question Number 194170 Answers: 0 Comments: 0
Question Number 194169 Answers: 0 Comments: 0
Question Number 194166 Answers: 1 Comments: 0
Question Number 194163 Answers: 0 Comments: 0
$$\boldsymbol{{Let}}\:\boldsymbol{{a}}\:,\:\boldsymbol{{b}}\:,\:\boldsymbol{{c}}\:\:\:\boldsymbol{{be}}\:\boldsymbol{{positive}}\:\boldsymbol{{real}}\:\boldsymbol{{numbers}} \\ $$$$\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\frac{\boldsymbol{{a}}}{\boldsymbol{{b}}}+\frac{\boldsymbol{{b}}}{\boldsymbol{{c}}}+\frac{\boldsymbol{{c}}}{\boldsymbol{{a}}}+\frac{\mathrm{3}^{\mathrm{3}} \sqrt{{abc}}}{\boldsymbol{{a}}+\boldsymbol{{b}}+\boldsymbol{{c}}}\geqslant\mathrm{4} \\ $$
Question Number 194159 Answers: 0 Comments: 0
$$...\mathrm{anybody}\:\mathrm{tried}\:\mathrm{question}\:\mathrm{193484}? \\ $$$$\mathrm{It}'\mathrm{s}\:\mathrm{not}\:\mathrm{as}\:\mathrm{hard}\:\mathrm{as}\:\mathrm{it}\:\mathrm{may}\:\mathrm{seem}. \\ $$
Question Number 194140 Answers: 1 Comments: 0
$${prove}\:{it}\:: \\ $$$$\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}^{\:\mathrm{2}} \geqslant\:\begin{pmatrix}{\:\:\:\:{n}}\\{{k}−\mathrm{1}}\end{pmatrix}\:×\begin{pmatrix}{\:\:{n}}\\{{k}+\mathrm{1}}\end{pmatrix}\:\:\:\:\:;\:\:\:\mathrm{1}\leqslant{k}\leqslant{n}−\mathrm{1} \\ $$$$ \\ $$
Question Number 194135 Answers: 3 Comments: 0
$$\:\:\:\:\:\:\begin{array}{|c|}{\frac{\mathrm{23}!−\mathrm{23}}{\mathrm{1}.\mathrm{1}!+\mathrm{2}.\mathrm{2}!+\mathrm{3}.\mathrm{3}!+...+\mathrm{21}.\mathrm{21}!}\:=?}\\\hline\end{array} \\ $$
Question Number 194113 Answers: 1 Comments: 0
Question Number 194105 Answers: 1 Comments: 0
$$ \\ $$$${x}\:,\:{y}\:,\:{z}\:{are}\:{positive}\:{real}\:{numbers}\:{if}\:{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} =\mathrm{1} \\ $$$${Then}\:{find}\:{the}\:{minimum}\:{value}\:{of}\: \\ $$$$\frac{{x}^{\mathrm{3}} }{\mathrm{1}−{x}^{\mathrm{8}} }+\frac{{y}^{\mathrm{3}} }{\mathrm{1}−{y}^{\mathrm{8}} }+\frac{{z}^{\mathrm{3}} }{\mathrm{1}−{z}^{\mathrm{8}} } \\ $$
Question Number 194086 Answers: 1 Comments: 1
$${f}\left({x}\right)=\mathrm{2}{y}^{\mathrm{2}} +\mathrm{4}{y}+\mathrm{3}\:\:{then}\:{a}_{\mathrm{0}\:} {equle}\:{to} \\ $$$$\left.{a}\left.\right)\:\:\:{p}\left(\mathrm{5}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{b}\right)\:\:\:\:\mathrm{7} \\ $$$$\left.{c}\left.\right)\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{d}\right)\:{not}\:{defined} \\ $$
Question Number 194065 Answers: 0 Comments: 0
Question Number 194057 Answers: 0 Comments: 0
$${find}\:{all}\:{functions}\:{such}\:{that} \\ $$$${f}\left({x}\right){f}\left({y}\right)\:=\:{x}^{{a}} {f}\left(\frac{{y}}{\mathrm{2}}\right)+{y}^{{b}} {f}\left(\frac{{x}}{\mathrm{2}}\right) \\ $$
Question Number 194037 Answers: 3 Comments: 0
$$ \\ $$$${Let}\:{a}_{\mathrm{1}} ,{a}_{\mathrm{2}} ....{a}_{{n}} \in{R}^{+} ,\:{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +.....{a}_{{n}} =\mathrm{1} \\ $$$${prove}\:{that}: \\ $$$$\frac{{a}_{\mathrm{1}} }{\mathrm{2}−{a}_{\mathrm{1}} }+\frac{{a}_{\mathrm{2}} }{\mathrm{2}−{a}_{\mathrm{2}} }.......\frac{{a}_{{n}} }{\mathrm{2}−{a}_{{n}} }\geqslant\frac{{n}}{\mathrm{2}{n}−\mathrm{1}} \\ $$
Question Number 193978 Answers: 1 Comments: 0
Question Number 193972 Answers: 0 Comments: 4
Question Number 193965 Answers: 2 Comments: 0
$${a},{b},{c}\:{are}\:{positive}\:{real}\:{numbers} \\ $$$${And} \\ $$$$\frac{\mathrm{1}}{{a}+{b}+\mathrm{1}}+\frac{\mathrm{1}}{{b}+{c}+\mathrm{1}}+\frac{\mathrm{1}}{{a}+{c}+\mathrm{1}}\geqslant\mathrm{1} \\ $$$${prove}\:{that}\:{a}+{b}+{c}\geqslant{ab}+{bc}+{ac} \\ $$
Question Number 193962 Answers: 2 Comments: 0
Question Number 194691 Answers: 0 Comments: 0
$${a},\:{b},\:{c}\geqslant\mathrm{0},\:{a}+{b}+{c}=\mathrm{2}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{3}{a}+\mathrm{8}{ab}+\mathrm{16}{abc}\leqslant\mathrm{12}. \\ $$
Question Number 193922 Answers: 4 Comments: 0
Question Number 193886 Answers: 2 Comments: 0
Question Number 193875 Answers: 2 Comments: 0
$${a},{b},{c},{d},{e},{f},\:{are}\:+\:{real}\:{numbers} \\ $$$${prove}: \\ $$$$\frac{{a}}{{b}+{c}}+\frac{{b}}{{c}+{d}}+\frac{{c}}{{d}+{e}}+\frac{{d}}{{e}+{f}}+\frac{{e}}{{f}+{a}}+\frac{{f}}{{a}+{b}}\geqslant\mathrm{3} \\ $$
Question Number 193848 Answers: 2 Comments: 0
Question Number 193796 Answers: 0 Comments: 0
$${Let}\:{G}\:{be}\:{a}\:{finite}\:{group},{f}\:{be}\:{an}\:{automorphism}\:{of}\:{G} \\ $$$${such}\:{that}\:{f}\left({x}\right)={x}\:\Rightarrow{x}={e}\:. \\ $$$${Then}\:{prove}\:{that}, \\ $$$$\left(\boldsymbol{{i}}\right)\forall{g}\in{G},\:\exists{x}\in{G}\:{such}\:{that}\:{g}={x}^{−\mathrm{1}} {f}\left({x}\right). \\ $$$$\left(\boldsymbol{{ii}}\right){If}\:\forall{x}\in{G}\:,\:{f}\left({f}\left({x}\right)\right)={x}\:\Rightarrow\:{G}\:{is}\:{Abelian}. \\ $$$$ \\ $$
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