Category: Algebra

Question Number 197706 by universe last updated on 26/Sep/23 Answered by witcher3 last updated on 27/Sep/23 $${\left(\frac{\sqrt{\mathrm{a}}+\sqrt{\mathrm{b}}}{2}\right)}^2⩽\frac{\mathrm{a}+\mathrm{b}}{2}\iff \frac{\mathrm{a}+\mathrm{b}}{4}⩾\frac{\sqrt{\mathrm{ab}}}{2}.\mathrm{Am}-\mathrm{GM}$$$$\left.\Rightarrow\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{y}^{\mathrm{2}} }}\leqslant\mathrm{2}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{y}^{\mathrm{2}} }\right.}\right) \…

Question Number 197618 by srijanGuha last updated on 24/Sep/23 Answered by Rasheed.Sindhi last updated on 24/Sep/23 $$3(x+\frac{1}{x})=2$$$$x+\frac{1}{x}=\frac{2}{3}$$$${(x+\frac{1}{x})}^3={\left(\frac{2}{3}\right)}^3$$$${x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}}…

Question Number 197567 by hardmath last updated on 21/Sep/23 Answered by Frix last updated on 22/Sep/23 $$\mathrm{Obviously}x=y=1$$ Commented by Frix last updated on…

Question Number 197476 by universe last updated on 19/Sep/23 $$$$$$\mathrm{find}\mathrm{the}\mathrm{sum}$$$$\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{4}{x^4+1}+\dots \dots \dots +\frac{2^n}{x^{2^n}+1}=??$$ Commented by witcher3 last…

Question Number 197464 by universe last updated on 18/Sep/23 Answered by witcher3 last updated on 19/Sep/23 $$\mathrm{claim}$$$$\frac{{(\mathrm{a}+\mathrm{b})}^6}{{\left(\mathrm{ab}\right)}^2}⩾32({\mathrm{a}}^2+{\mathrm{b}}^2)\dots .\mathrm{P},\mathrm{by}\mathrm{symetri}\mathrm{a}⩾\mathrm{b}$$$$\left.\mathrm{a}\left.=\mathrm{tb},\mathrm{t}\in\right]\mathrm{0},\mathrm{1}\right]…