Question Number 205654 by Lindemann last updated on 26/Mar/24 $${We}\:{define}\:{a}\:{domino}\:{as}\:{being}\:{and}\:{ordered}\:{pair}\:{of}\:{distinct} \\ $$$${integers}.\:{A}\:{suitable}\:{sequence}\:{of}\:{dominos}\:{is}\:{a}\:{list}\:{of}\:{distinct} \\ $$$${dominoes}\:{in}\:{which}\:{the}\:{first}\:{coordonate}\:{of}\:{each}\:{pair}\:{after} \\ $$$${the}\:{first}\:{is}\:{equal}\:{to}\:{the}\:{second}\:{coordonate}\:{of}\:{the}\:{immediately} \\ $$$${preceding}\:{pair},\:{and}\:{in}\:{which}\:{the}\:{pairs}\:\left({i};{j}\right)\:{and}\:\left({j};{i}\right) \\ $$$${do}\:{not}\:{both}\:{appear}\:{for}\:{all}\:{i}\:{and}\:{j}.\:{Let}\:{D}_{\mathrm{40}} \:{the} \\ $$$${set}\:{of}\:{all}\:{dominoes}\:{whose}\:{coordonate}\:{are}\:{not}\:{greater} \\ $$$${than}\:\mathrm{40}.\:{Find}\:{the}\:{length}\:{of}\:{the}\:{longest}\:{suitable}\:{sequence}…
Question Number 205645 by hardmath last updated on 26/Mar/24 $$\mathrm{Find}:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \:\frac{\mathrm{sin}^{\mathrm{2}} \mathrm{x}}{\mathrm{2}\:\mathrm{cosx}\:+\:\mathrm{3}\:\mathrm{sinx}}\:\mathrm{dx}\:=\:? \\ $$ Answered by Frix last updated on 26/Mar/24 $${t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\:\mathrm{leads}\:\mathrm{to} \\ $$$$−\mathrm{4}\int\frac{{t}^{\mathrm{2}}…
Question Number 205672 by hardmath last updated on 26/Mar/24 Answered by Berbere last updated on 27/Mar/24 $$\Omega=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{\sqrt{{k}\left({k}−\mathrm{1}\right)}}{\:\sqrt{\left(\sqrt{{k}−\mathrm{1}}+\sqrt{{k}}\right)^{\mathrm{2}} }}.\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\sqrt{{k}}\right)^{−\mathrm{1}} \\ $$$${S}_{\mathrm{1}}…
Question Number 205640 by hardmath last updated on 26/Mar/24 $$\mathrm{If}\:\:\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{abc}\geqslant\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\geqslant\:\frac{\mathrm{1}+\mathrm{a}}{\mathrm{1}+\mathrm{b}}\:+\:\frac{\mathrm{1}+\mathrm{b}}{\mathrm{1}+\mathrm{c}}\:+\:\frac{\mathrm{1}+\mathrm{c}}{\mathrm{1}+\mathrm{a}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 205643 by hardmath last updated on 26/Mar/24 $$\mathrm{If}\:\:\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:=\:\mathrm{abc} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{a}}{\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{bc}}\:+\:\frac{\mathrm{b}}{\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{ac}}\:+\:\frac{\mathrm{c}}{\mathrm{c}^{\mathrm{2}} \:+\:\mathrm{ab}}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$ Answered by A5T…
Question Number 205669 by hardmath last updated on 26/Mar/24 Answered by Berbere last updated on 27/Mar/24 $${A}={X}^{\mathrm{5}} \\ $$$$\chi_{{A}} =\left(\mathrm{3}−{x}\right)\left(\mathrm{5}−{x}\right)−\mathrm{8}={X}^{\mathrm{2}} −\mathrm{8}{X}+\mathrm{7} \\ $$$$\left({x}−\mathrm{1}\right)\left({X}−\mathrm{7}\right)\:\:{D}=\begin{pmatrix}{\mathrm{1}\:\:\mathrm{0}}\\{\mathrm{0}\:\:\mathrm{7}}\end{pmatrix} \\ $$$${U}_{\mathrm{1}}…
Question Number 205670 by hardmath last updated on 26/Mar/24 Answered by Berbere last updated on 27/Mar/24 $$\Omega=\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \left(\frac{\mathrm{tan}^{−\mathrm{1}} \left({x}\right)}{{x}−\mathrm{tan}^{−\mathrm{1}} \left({x}\right)}\right)^{\mathrm{2}} {dx} \\ $$$$\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}}…
Question Number 205671 by hardmath last updated on 26/Mar/24 Answered by Berbere last updated on 27/Mar/24 $$\left(\mathrm{1}−{x}\right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} {x}^{{k}} \begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\Rightarrow\int_{\mathrm{0}} ^{{t}} \underset{{k}=\mathrm{0}} {\overset{{n}}…
Question Number 205639 by mr W last updated on 26/Mar/24 Answered by mahdipoor last updated on 26/Mar/24 $${Trapezium}−\mathrm{2}\:{Right}\:{triangle}={area}\:\Rightarrow \\ $$$$\left[\frac{\left({r}\right)+\left(\mathrm{4}+{r}\right)}{\mathrm{2}}×\mathrm{4}\right]−\left[\frac{\left(\mathrm{4}+{r}\right)\left({r}\right)}{\mathrm{2}}+\frac{\left(\mathrm{4}−{r}\right)\left({r}\right)}{\mathrm{2}}\right]=\mathrm{8} \\ $$ Commented by mr…
Question Number 205596 by cortano12 last updated on 25/Mar/24 Commented by A5T last updated on 25/Mar/24 $${DE}\:{is}\:{not}\:{uniquely}\:{defined}. \\ $$ Answered by TheHoneyCat last updated on…