Question Number 5380 by 314159 last updated on 12/May/16 $${Suppose}\:{that}\:{a},{b},{c}>\mathrm{0}.{Prove}\:{that}\: \\ $$$$\frac{\mathrm{1}}{{a}\left(\mathrm{1}+{b}\right)}+\frac{\mathrm{1}}{{b}\left(\mathrm{1}+{c}\right)}+\frac{\mathrm{1}}{{c}\left(\mathrm{1}+{a}\right)}\:\geqslant\frac{\mathrm{3}}{\mathrm{1}+{abc}}. \\ $$ Commented by Rasheed Soomro last updated on 14/May/16 $$\mathrm{LHS}=\frac{{bc}\left(\mathrm{1}+{c}\right)\left(\mathrm{1}+{a}\right)+{ac}\left(\mathrm{1}+{b}\right)\left(\mathrm{1}+{a}\right)+{ab}\left(\mathrm{1}+{b}\right)\left(\mathrm{1}+{c}\right)}{{a}\mathrm{bc}\left(\mathrm{1}+\mathrm{a}\right)\left(\mathrm{1}+{b}\right)\left(\mathrm{1}+\mathrm{c}\right)} \\ $$$$=\frac{{ab}\left(\mathrm{1}+{c}+{b}+{bc}\right)+{bc}\left(\mathrm{1}+{a}+{c}+{ca}\right)+{ca}\left(\mathrm{1}+{a}+{b}+{ab}\right)}{{a}\mathrm{bc}\left(\mathrm{1}+\mathrm{a}\right)\left(\mathrm{1}+{b}\right)\left(\mathrm{1}+\mathrm{c}\right)}…
Question Number 70915 by Mr. K last updated on 09/Oct/19 Commented by Mr. K last updated on 09/Oct/19 $${The}\:{circles}\:{have}\:{the}\:{same}\:{radius}.\: \\ $$$${The}\:{triangle}\:{is}\:{equilateral}\:{side} \\ $$$$\mathrm{28}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right).\:{Determine}\:{the}\:{radius} \\ $$$${of}\:{the}\:{circumferences}.…
Question Number 5375 by Junaid Mirza last updated on 12/May/16 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 5357 by Junaid Mirza last updated on 11/May/16 Commented by Yozzii last updated on 11/May/16 $${Let}\:{the}\:{lowest}\:{position}\:{of}\:{the}\:{bob}\:{be} \\ $$$${that}\:{where}\:{its}\:{gravitational}\:{potential} \\ $$$${energy}\:{is}\:{zero}.\:{Then},\:{by}\:{the}\:{law}\:{of} \\ $$$${conservation}\:{of}\:{energy}\:{for}\:{the}\:{bob}, \\…
Question Number 70887 by naka3546 last updated on 09/Oct/19 Commented by ajfour last updated on 09/Oct/19 Commented by ajfour last updated on 09/Oct/19 $${s}\mathrm{sin}\:\alpha={r}\mathrm{sin}\:\gamma \\…
Question Number 70885 by naka3546 last updated on 09/Oct/19 $$\sqrt[{\mathrm{3}}]{\mathrm{2}}\:\:=\:{a}\:+\:\frac{\mathrm{1}}{{b}\:+\:\frac{\mathrm{1}}{{c}\:+\:\frac{\mathrm{1}}{{d}\:+\:\ldots}}} \\ $$$${a},\:{b},\:{c},\:{d}\:\:\in\:\mathbb{Z}^{+} \\ $$$${What}'{s}\:\:{b}\:\:? \\ $$ Answered by MJS last updated on 09/Oct/19 $$\mathrm{the}\:\mathrm{continued}\:\mathrm{fraction}\:\mathrm{of}\:\sqrt[{\mathrm{3}}]{\mathrm{2}}\:\mathrm{is}\:\mathrm{non}+\mathrm{periodic} \\…
Question Number 70874 by Mr. K last updated on 09/Oct/19 $${If}\:\mathrm{4}−\mathrm{2}\sqrt{\mathrm{5}}\:{and}\:\mathrm{4}+\mathrm{2}\sqrt{\mathrm{5}\:}\:{are}\:{solutions} \\ $$$${of}\:{x}^{\mathrm{2}} +\left(\mathrm{5}{a}−{b}\right){x}+\left(\mathrm{3}{b}−{a}\right)=\mathrm{0} \\ $$$${whete}\:{a}\:{and}\:{b}\:{are}\:{real}\:{numbers},\: \\ $$$${determine}\:{the}\:{product}\:{of}\:\boldsymbol{{ab}}. \\ $$ Answered by tw000001 last updated…
Question Number 5337 by Junaid Mirza last updated on 09/May/16 Commented by prakash jain last updated on 09/May/16 $$\mathrm{4}\:\mathrm{is}\:\mathrm{correct}. \\ $$$$\mathrm{Relative}\:\mathrm{refractive}\:\mathrm{index}\:\:_{\mathrm{1}} \mu_{\mathrm{2}} =\frac{\mu_{\mathrm{2}} }{\mu_{\mathrm{1}} }…
Question Number 5336 by Junaid Mirza last updated on 09/May/16 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 70847 by A8;15: last updated on 08/Oct/19 Answered by mind is power last updated on 08/Oct/19 $$\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2}}\\{\mathrm{2}{x}^{\mathrm{4}} +{xy}^{\mathrm{3}} +{x}^{\mathrm{2}} {y}^{\mathrm{2}} −{x}^{\mathrm{3}}…