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Category: Arithmetic

A-triangle-is-formed-by-the-three-straight-line-y-m-1-x-a-m-1-y-m-2-x-a-m-2-y-m-3-x-a-m-3-prove-that-its-orthocenter-always-lies-on-the-line-x-a-0-

Question Number 76717 by peter frank last updated on 29/Dec/19 $${A}\:{triangle}\:{is}\:{formed}\:{by} \\ $$$${the}\:{three}\:{straight}\:{line} \\ $$$${y}={m}_{\mathrm{1}} {x}+\frac{{a}}{{m}_{\mathrm{1}} } \\ $$$${y}={m}_{\mathrm{2}} {x}+\frac{{a}}{{m}_{\mathrm{2}} } \\ $$$${y}={m}_{\mathrm{3}} {x}+\frac{{a}}{{m}_{\mathrm{3}} }…

If-u-arcsin-x-y-arctan-y-x-show-that-x-u-dx-y-u-dy-0-

Question Number 76715 by peter frank last updated on 29/Dec/19 $${If}\:{u}={arc}\mathrm{sin}\:\frac{{x}}{{y}}+{arc}\mathrm{tan}\:\frac{{y}}{{x}} \\ $$$${show}\:{that}\: \\ $$$${x}\frac{\partial{u}}{{dx}}+{y}\frac{\partial{u}}{{dy}}=\mathrm{0} \\ $$ Commented by abdomathmax last updated on 30/Dec/19 $$\frac{\partial{u}}{\partial{x}}\:=\frac{\mathrm{1}}{{y}\sqrt{\mathrm{1}−\frac{{x}^{\mathrm{2}}…

n-1-1-1-n-4-

Question Number 142150 by iloveisrael last updated on 27/May/21 $$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\right)\:=?\: \\ $$ Commented by liberty last updated on 27/May/21 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}}…

prove-that-9-4-lt-log-2-3-2-lt-25-9-

Question Number 11005 by ajfour last updated on 06/Mar/17 $${prove}\:{that}\:\frac{\mathrm{9}}{\mathrm{4}}\:<\:\left(\mathrm{log}\:_{\mathrm{2}} \mathrm{3}\right)^{\mathrm{2}} \:<\:\frac{\mathrm{25}}{\mathrm{9}}\:\:. \\ $$ Answered by mrW1 last updated on 06/Mar/17 $$\mathrm{2log}_{\mathrm{2}} \:\mathrm{3}=\mathrm{log}_{\mathrm{2}} \:\mathrm{3}^{\mathrm{2}} =\mathrm{log}_{\mathrm{2}}…