Question Number 178354 by Spillover last updated on 15/Oct/22 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{sinh}\:\mathrm{x}+\mathrm{ncosh}\:\mathrm{x}\:\mathrm{is}\:\sqrt{\mathrm{n}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\mathrm{and}\:\mathrm{this}\:\mathrm{occurs}\:\mathrm{x}=\mathrm{0}.\mathrm{5ln}\:\left(\frac{\mathrm{n}−\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right) \\ $$ Answered by Ar Brandon last updated on 16/Oct/22…
Question Number 178347 by Acem last updated on 15/Oct/22 $${Let}\:{f}\left({x}\right)=\:\left({ax}+\mathrm{1}\right)^{\mathrm{5}} .\left(\mathrm{1}+{bx}\right)^{\mathrm{4}} \:;\:{a},{b}\:\in\:\mathbb{N} \\ $$$$\:{if}\:{times}\:{of}\:{x}\:{equal}\:\mathrm{62}\:{so}\:{what}\:{are}\:{possible}\:{values} \\ $$$$\:{of}\:{the}\:{sum}\:{a},\:{b}? \\ $$$$ \\ $$ Answered by Acem last updated…
Question Number 112804 by Aina Samuel Temidayo last updated on 09/Sep/20 $$\mathrm{If}\:\frac{\mathrm{97}}{\mathrm{19}}=\:\mathrm{a}+\frac{\mathrm{1}}{\mathrm{b}+\frac{\mathrm{1}}{\mathrm{c}}},\:\mathrm{where}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{are} \\ $$$$\mathrm{positive}\:\mathrm{integers},\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum} \\ $$$$\mathrm{of}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}? \\ $$ Answered by nimnim last updated on 09/Sep/20…
Question Number 112802 by Aina Samuel Temidayo last updated on 09/Sep/20 $$\mathrm{If}\:\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=\mathrm{0},\:\mathrm{then} \\ $$$$ \\ $$$$\left(\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} −\mathrm{yz}}+\frac{\mathrm{1}}{\mathrm{y}^{\mathrm{2}} −\mathrm{zx}}+\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{2}} −\mathrm{xy}}\right)\left(\mathrm{x},\mathrm{y},\mathrm{z}\:\neq\:\mathrm{0}\right)\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$$$ \\ $$…
Question Number 112805 by Aina Samuel Temidayo last updated on 09/Sep/20 $$\mathrm{If}\:\mathrm{x}=\frac{\sqrt{\mathrm{a}+\mathrm{2b}}+\:\sqrt{\mathrm{a}−\mathrm{2b}}}{\:\sqrt{\mathrm{a}+\mathrm{2b}}−\:\sqrt{\mathrm{a}−\mathrm{2b}}},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{bx}^{\mathrm{2}} −\mathrm{ax}+\mathrm{b}\:\mathrm{is} \\ $$ Answered by nimnim last updated on 09/Sep/20 $$\:\:\:\:\:\frac{\mathrm{x}}{\mathrm{1}}=\frac{\sqrt{\mathrm{a}+\mathrm{2b}}+\sqrt{\mathrm{a}−\mathrm{2b}}}{\:\sqrt{\mathrm{a}+\mathrm{2b}}−\sqrt{\mathrm{a}−\mathrm{2b}}}…
Question Number 112803 by Aina Samuel Temidayo last updated on 09/Sep/20 $$\mathrm{If}\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{0},\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} −\mathrm{yz}}\:\mathrm{is} \\ $$ Answered by MJS_new last updated…
Question Number 178338 by Acem last updated on 15/Oct/22 $${Let}\:{S}=\:\left\{\mathrm{1},\:\mathrm{2},\:…,\:\mathrm{9},\:\mathrm{0}\right\}\:\boldsymbol{{A}}:\:{How}\:{many}\:{multiples} \\ $$$$\:{of}\:{eight}\:{of}\:{four}\:{digits}\:{can}\:{be}\:{formed}\:{from}\:{S} \\ $$$$\:\boldsymbol{{B}}:\:{The}\:{same}\:{question}\:{for}\:{different}\:{digits} \\ $$ Commented by Acem last updated on 17/Oct/22 $$\boldsymbol{{Hint}}: \\…
Question Number 112796 by nimnim last updated on 09/Sep/20 $${For}\:{any}\:{real}\:{numbers}\:{x},{y}\:{and}\:{z}\:{if} \\ $$$${x}+{y}+{z}=\mathrm{2},\:{then}\:{prove}\:{that} \\ $$$${xyz}\geqslant\mathrm{8}\left(\mathrm{1}−{x}\right)\left(\mathrm{1}−{y}\right)\left(\mathrm{1}−{z}\right). \\ $$ Commented by nimnim last updated on 10/Sep/20 $$\mathrm{Yes}\:\mathrm{sir}.. \\…
Question Number 112795 by I want to learn more last updated on 09/Sep/20 Commented by MJS_new last updated on 10/Sep/20 $$\mathrm{trying}\:\mathrm{10}\leqslant{N}\leqslant\mathrm{99}\:\mathrm{we}\:\mathrm{get} \\ $$$${N}\in\left\{\mathrm{12},\:\mathrm{24},\:\mathrm{36},\:\mathrm{48},\:\mathrm{75},\:\mathrm{81}\right\} \\ $$…
Question Number 178320 by Shrinava last updated on 15/Oct/22 Commented by mr W last updated on 15/Oct/22 $${only}\:{if}\:\mathrm{log}\:\mathrm{3}\:{means}\:\mathrm{log}_{{e}} \:\mathrm{3},\:{i}.{e}.\:\mathrm{ln}\:\mathrm{3} \\ $$ Commented by Acem last…