Question Number 20298 by mondodotto@gmail.com last updated on 25/Aug/17 Answered by Tinkutara last updated on 25/Aug/17 $$\mathrm{45}\:=\:\mathrm{36}\:+\:\mathrm{12}\:+\:\mathrm{18}\:−\:{n}\left(\mathrm{medals}\:\mathrm{in}\:\mathrm{two}\right. \\ $$$$\left.\mathrm{categories}\right)\:+\:\mathrm{4} \\ $$$${n}\left(\mathrm{medals}\:\mathrm{in}\:\mathrm{two}\:\mathrm{categories}\right)\:=\:\mathrm{25} \\ $$$${n}\left(\mathrm{exactly}\:\mathrm{two}\:\mathrm{categories}\right)\:=\:\mathrm{25}−\mathrm{3}×\mathrm{4} \\ $$$$=\:\mathrm{13}…
Question Number 20296 by Tinkutara last updated on 25/Aug/17 $$\mathrm{If}\:\left({m}_{{r}} \:,\:\frac{\mathrm{1}}{{m}_{{r}} }\right)\:;\:{r}\:=\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\:\mathrm{be}\:\mathrm{four}\:\mathrm{pairs} \\ $$$$\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{and}\:{y}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{gx}\:+\:\mathrm{2}{fy}\:+\:{c}\:=\:\mathrm{0},\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:{m}_{\mathrm{1}} .{m}_{\mathrm{2}} .{m}_{\mathrm{3}} .{m}_{\mathrm{4}} \:=\:\mathrm{1}. \\…
Question Number 20297 by Tinkutara last updated on 25/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{expression}\:{ax}^{\mathrm{2}} \:+\:\mathrm{2}{hxy} \\ $$$$+\:{by}^{\mathrm{2}} \:+\:\mathrm{2}{gx}\:+\:\mathrm{2}{fy}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{resolved}\:\mathrm{into}\:\mathrm{two}\:\mathrm{linear}\:\mathrm{rational}\:\mathrm{factors} \\ $$$$\mathrm{if}\:\Delta\:=\:{abc}\:+\:\mathrm{2}{fgh}\:−\:{af}^{\mathrm{2}} \:−\:{bg}^{\mathrm{2}} \:−\:{ch}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$ Answered by…
Question Number 151361 by mathdanisur last updated on 20/Aug/21 $$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}=\mathrm{45}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}\:-\:\mathrm{tan}\left(\mathrm{x}\right)}{\mathrm{1}\:-\:\mathrm{tan}\left(\mathrm{y}\right)}\:=\:\mathrm{tan}\left(\mathrm{2y}\right) \\ $$ Answered by Ar Brandon last updated on 20/Aug/21 $${x}+{y}=\frac{\pi}{\mathrm{4}} \\ $$$$\frac{\mathrm{1}−\mathrm{tan}{x}}{\mathrm{1}−\mathrm{tan}{y}}=\frac{\mathrm{1}−\mathrm{tan}\left(\frac{\pi}{\mathrm{4}}−{y}\right)}{\mathrm{1}−\mathrm{tan}{y}}…
Question Number 85822 by jagoll last updated on 25/Mar/20 $$\mathrm{how}\:\mathrm{to}\:\mathrm{solve}\: \\ $$$$\sqrt[{\mathrm{3}\:\:}]{\mathrm{x}−\mathrm{1}}\:+\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{x}−\mathrm{3}}\:+\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{x}−\mathrm{5}}\:=\:\mathrm{0}\: \\ $$ Commented by jagoll last updated on 25/Mar/20 $$\mathrm{thank}\:\mathrm{mr}\: \\ $$ Answered…
Question Number 151357 by mathdanisur last updated on 20/Aug/21 Commented by mr W last updated on 20/Aug/21 $${check}\:{the}\:{question}\:{please}! \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}{a}_{{k}} \geqslant\mathrm{100}\:{and}\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}{a}_{{k}}…
Question Number 151350 by mathdanisur last updated on 20/Aug/21 Commented by dumitrel last updated on 20/Aug/21 $$\mathrm{0}\leqslant\lambda\leqslant\mathrm{8}\:{sau}\:\mathrm{0}\leqslant\lambda\leqslant\mathrm{5}\:????? \\ $$$${p}^{\mathrm{2}} \geqslant\mathrm{3}{q}\Rightarrow{q}\leqslant\mathrm{3} \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{3}} ={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}}…
Question Number 151351 by mathdanisur last updated on 20/Aug/21 Answered by dumitrel last updated on 20/Aug/21 $$\:\lambda\sqrt{\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{\mathrm{2}}}+\frac{\mathrm{2}{ab}}{{a}+{b}}\geqslant\frac{\lambda+\mathrm{1}}{\mathrm{2}}\left({a}+{b}\right) \\ $$$${if}\:{a}={b}\:{true} \\ $$$${suppose}\:{a}<{b}\:;\frac{{a}}{{b}}={t}\in\left(\mathrm{0};\mathrm{1}\right) \\ $$$$\Leftrightarrow\lambda\sqrt{\frac{{t}^{\mathrm{2}}…
Question Number 151346 by mathdanisur last updated on 20/Aug/21 $$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{5x}\:+\:\mathrm{25}\:=\:\mathrm{0}\:\:\Rightarrow\:\:\mathrm{x}^{\mathrm{3}} \:-\:\mathrm{100}\:=\:? \\ $$ Answered by Rasheed.Sindhi last updated on 20/Aug/21 $$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{5x}\:+\:\mathrm{25}\:=\:\mathrm{0}\:\:\Rightarrow\:\:\mathrm{x}^{\mathrm{3}} \:-\:\mathrm{100}\:=\:? \\…
Question Number 151344 by mathdanisur last updated on 20/Aug/21 $$\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{4x}\:+\:\mathrm{1}\:=\:\mathrm{0}\:\:\Rightarrow\:\:\mathrm{x}^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} }\:=\:? \\ $$ Answered by dumitrel last updated on 20/Aug/21 $${x}−\mathrm{4}+\frac{\mathrm{1}}{{x}}=\mathrm{0}\Rightarrow{x}+\frac{\mathrm{1}}{{x}}=\mathrm{4}\Rightarrow \\ $$$$\mathrm{64}={x}^{\mathrm{3}}…