Question Number 218644 by Spillover last updated on 13/Apr/25 Answered by Spillover last updated on 14/Apr/25 Answered by Spillover last updated on 15/Apr/25 Terms of…
Question Number 218645 by Spillover last updated on 13/Apr/25 Answered by Spillover last updated on 14/Apr/25 Answered by Spillover last updated on 14/Apr/25 Answered by…
Question Number 218646 by Spillover last updated on 13/Apr/25 Commented by Spillover last updated on 13/Apr/25 $${what}\:{fraction}\:{of}\:{the} \\ $$$${square}\:{is}\:{shaded} \\ $$ Answered by Spillover last…
Question Number 218609 by MrGaster last updated on 13/Apr/25 $$\mathrm{Can}\:{y}={x}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{as}\:\mathrm{them} \\ $$$$\mathrm{su}\:\mathrm{of}\:\mathrm{two}\:\mathrm{periodic}\:\mathrm{functions}? \\ $$ Answered by MrGaster last updated on 13/Apr/25 $$\mathbb{R}\in{x}={f}\left({x}\right)+{g}\left({x}\right)\:\exists{T}_{\mathrm{1}\:} ,{T}\mathrm{2}>\mathrm{0}:{f}\left({x}+{T}_{\mathrm{1}} \right)−{f}\left({x}\right)\wedge{g}\left({x}+{T}_{\mathrm{2}\:} \right)={g}\left({x}\right)\:\forall{x}\in\mathbb{R}…
Question Number 218636 by hardmath last updated on 13/Apr/25 Answered by vnm last updated on 14/Apr/25 $${let} \\ $$$$\:{a}_{{k}} =\frac{\mathrm{1}}{\mathrm{3}{k}−\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{3}{k}−\mathrm{2}}−\frac{\mathrm{2}}{\mathrm{3}{k}} \\ $$$${b}_{{k}} =\frac{\mathrm{4}{k}}{\mathrm{4}{k}+\mathrm{1}}−\frac{\mathrm{4}\left({k}−\mathrm{1}\right)}{\mathrm{4}\left({k}−\mathrm{1}\right)+\mathrm{1}} \\ $$$${if}\:\:\forall{k}\geqslant\mathrm{1}\:{a}_{{k}}…
Question Number 218629 by galisamikshareddy last updated on 13/Apr/25 Answered by galisamikshareddy last updated on 13/Apr/25 $${no}.{of}\:{hours}\:{rehana}\:{works}\:{each}\:{day}\: \\ $$$$=\mathrm{3}\frac{\mathrm{1}}{\mathrm{2}}\:{hours}\: \\ $$$$=\frac{\mathrm{5}}{\mathrm{2}}\:{hours} \\ $$$${no}.{of}\:{days}\:{she}\:{completed}\:{her}\:{work}\: \\ $$$$=\:\mathrm{7}{days}…
Question Number 218624 by Nicholas666 last updated on 13/Apr/25 $$ \\ $$$$\:\:\:\:{Prove};\:\underset{\frac{\mathrm{1}}{{e}}} {\int}^{{e}} \:\frac{{t}^{\mathrm{2}} }{{e}^{{t}^{\mathrm{2}\:} } }\:{dt}\:\leqslant\:\mathrm{1}−\:\frac{\mathrm{1}}{{e}^{\mathrm{2}\:\:} } \\ $$$$\:\:\:{e}−{the}\:{base}\:{of}\:{natural}\:{logarithm} \\ $$$$ \\ $$ Answered…
Question Number 218626 by Nicholas666 last updated on 13/Apr/25 $$ \\ $$$$\:{Prove}\:{that}\:{for}\:{all}\:{real}\:{numbers}\:{a}\:{and}\:{b} \\ $$$${with}\:{a}<{b},\:{the}\:{following}\:{inequality}\:{holds}; \\ $$$$\left(\int_{{a}} ^{{b}} \mathrm{1}\:{dx}\right)^{\mathrm{3}} \leqslant\:\left({b}−{a}\right)\left(\int_{{a}} ^{{b}} \left({x}−{a}+\mathrm{1}\right)^{\mathrm{2}} {dx}\right)\left(\int_{{a}\:} ^{{b}} \frac{\mathrm{1}}{\left({a}−{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}\right)…
Question Number 218557 by efronzo1 last updated on 12/Apr/25 $$\: \\ $$ Commented by efronzo1 last updated on 12/Apr/25 Answered by A5T last updated on…
Question Number 218580 by mr W last updated on 12/Apr/25 Commented by vnm last updated on 12/Apr/25 $$ \\ $$$$\mathrm{Let}\:\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{be}\:\mathrm{inscribed}\:\mathrm{in}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{DEF}. \\ $$$$\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{DEF}\:\mathrm{is}\:\mathrm{maximal}\: \\ $$$$\mathrm{if}\:\mathrm{there}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{point}\:\mathrm{G}\:\mathrm{inside}\:\mathrm{ABC}\:\mathrm{such}\:\mathrm{that}\:\mathrm{GA},\:\mathrm{GB},\:\mathrm{GC}\:\mathrm{are}\:\mathrm{perpendicular}\: \\…