Question Number 110608 by Engr_Jidda last updated on 29/Aug/20 $$\sqrt{{x}}\:+\mathrm{1}=\mathrm{0} \\ $$$${Solution} \\ $$$$\sqrt{{x}\:}\:=\:−\mathrm{1} \\ $$$${recalled}\:{that}\:\varrho^{{i}\Pi} =\:−\mathrm{1} \\ $$$$\sqrt{{x}}\:=\varrho^{{i}\Pi} \\ $$$${x}=\left(\varrho^{{i}\Pi} \right)^{\mathrm{2}} \\ $$$$\Rightarrow\:{x}=\varrho^{\mathrm{2}{i}\Pi} \\…
Question Number 176141 by Linton last updated on 13/Sep/22 $${which}\:{is}\:{greater}\:? \\ $$$$\mathrm{4}^{\sqrt{\mathrm{2}}} \:{or}\:\mathrm{3}^{\sqrt{\mathrm{3}}} \\ $$ Commented by Linton last updated on 13/Sep/22 $${without}\:{calculator}\:{show}\:{working} \\ $$…
Question Number 110596 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{Three}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{satisfying} \\ $$$$\mathrm{ab}+\mathrm{c}=\mathrm{10},\mathrm{bc}+\mathrm{a}=\mathrm{11},\mathrm{ca}+\mathrm{b}=\mathrm{14}.\:\mathrm{Find} \\ $$$$\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{b}−\mathrm{c}\right)\left(\mathrm{c}−\mathrm{a}\right)\left(\mathrm{a}−\mathrm{1}\right)\left(\mathrm{b}−\mathrm{1}\right)\left(\mathrm{c}−\mathrm{1}\right) \\ $$ Commented by Her_Majesty last updated on 29/Aug/20…
Question Number 176128 by Shrinava last updated on 13/Sep/22 $$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:,\:\mathrm{cot}\:\frac{\mathrm{A}}{\mathrm{2}}\:,\:\mathrm{cot}\:\frac{\mathrm{B}}{\mathrm{2}}\:,\:\mathrm{cot}\:\frac{\mathrm{C}}{\mathrm{2}}\:\in\:\mathrm{Q} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\left(\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{n}}} +\:\left(\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\mathrm{cos}\:\frac{\mathrm{A}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{n}}} \in\:\mathrm{Q}\:,\:\forall\mathrm{n}\in\mathbb{N} \\ $$ Commented by Rasheed.Sindhi last updated…
Question Number 110594 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{A}\:\mathrm{polynomial}\:\mathrm{satisfies} \\ $$$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2}\right)=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(−\mathrm{x}\right).\:\mathrm{Assuming}\:\mathrm{that} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\neq\mathrm{0}\:\mathrm{for}\:−\mathrm{2}\leqslant\mathrm{x}\leqslant\mathrm{2},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{f}\left(−\mathrm{2}\right)+\mathrm{f}\left(\mathrm{1}\right)? \\ $$ Answered by Aziztisffola last…
Question Number 110592 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}=\frac{\mathrm{1}}{\mathrm{32}} \\ $$ Answered by Aziztisffola last updated on 29/Aug/20 $$\:\mathrm{if}\:\mathrm{x}=\mathrm{y}\:\Rightarrow\:\frac{\mathrm{2}}{\mathrm{x}}=\frac{\mathrm{1}}{\mathrm{32}}\Rightarrow\mathrm{x}=\mathrm{64} \\…
Question Number 110593 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{integer}\:\mathrm{n} \\ $$$$\mathrm{such}\:\mathrm{that}\:\frac{\left(\mathrm{n}−\mathrm{1}\right)\left(\mathrm{n}^{\mathrm{2}} +\mathrm{n}−\mathrm{3}\right)}{\mathrm{n}^{\mathrm{2}} +\mathrm{4}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{an} \\ $$$$\mathrm{integer} \\ $$ Answered by floor(10²Eta[1]) last updated…
Question Number 110586 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{solution} \\ $$$$\mathrm{a}\leqslant\mathrm{2x}+\mathrm{5}\leqslant\mathrm{b}\:\mathrm{is}\:\mathrm{15}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{b}−\mathrm{a}? \\ $$ Answered by Aziztisffola last updated on 29/Aug/20 $$\:\frac{\mathrm{a}−\mathrm{5}}{\mathrm{2}}\leqslant\mathrm{x}\leqslant\frac{\mathrm{b}−\mathrm{5}}{\mathrm{2}} \\…
Question Number 110587 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{A}\:\mathrm{palindrome}\:\mathrm{is}\:\mathrm{a}\:\mathrm{number}\:\mathrm{that} \\ $$$$\mathrm{remains}\:\mathrm{the}\:\mathrm{same}\:\mathrm{when}\:\mathrm{its}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{reversed}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{n}\:\mathrm{and} \\ $$$$\mathrm{n}+\mathrm{192}\:\mathrm{are}\:\mathrm{three}−\mathrm{digit}\:\mathrm{and} \\ $$$$\mathrm{four}−\mathrm{digit}\:\mathrm{palindromes}\:\mathrm{respectively}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{m}?\: \\ $$$$ \\…
Question Number 176101 by BaliramKumar last updated on 12/Sep/22 $${x}^{\mathrm{2}} \:−\:\mathrm{4}{y}^{\mathrm{2}} \:=\:\mathrm{3}{z}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\left[\:{x},\:{y},\:{z}\:\in\:\mathbb{N}\:\right]\: \\ $$$$\:\:\:\:\:\:\:\:\:\:{HCF}\left({x},\:{y},\:{z}\right)\:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{like}\:\left(\mathrm{4},\:\mathrm{1},\:\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x},\:{y},\:{z}\:<\:\mathrm{100} \\ $$$${solve}\:{by}\:{computer}\:{programing} \\ $$ Commented by Ar Brandon…