Question Number 213000 by golsendro last updated on 28/Oct/24 $$\:\:\:\mathrm{f}\left(\frac{\mathrm{x}−\mathrm{3}}{\mathrm{x}+\mathrm{1}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{x}+\mathrm{3}}{\mathrm{1}−\mathrm{x}}\right)\:=\:\mathrm{x}\:,\:\mathrm{x}\neq\:\pm\:\mathrm{1} \\ $$$$\:\:\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$ Answered by Ghisom last updated on 28/Oct/24 $$\frac{{u}−\mathrm{3}}{{u}+\mathrm{1}}={v}\:\Leftrightarrow\:{u}=\frac{{v}+\mathrm{3}}{\mathrm{1}−{v}} \\ $$$$\mathrm{2}{f}\left({x}\right)=\frac{{x}−\mathrm{3}}{{x}+\mathrm{1}}+\frac{{x}+\mathrm{3}}{\mathrm{1}−{x}}−{x} \\…
Question Number 213001 by efronzo1 last updated on 28/Oct/24 $$\:\:\:\:\:\:\underline{\underbrace{\lessdot}\cancel{} } \mathrm{14}^{\mathrm{3}} +\mathrm{15}^{\mathrm{3}} +\mathrm{16}^{\mathrm{3}} +…+\mathrm{24}^{\mathrm{3}} +\mathrm{25}^{\mathrm{3}} \\ $$ Commented by Ghisom last updated on 28/Oct/24…
Question Number 213002 by efronzo1 last updated on 28/Oct/24 $$\:\:\frac{\left(\mathrm{52}+\mathrm{6}\sqrt{\mathrm{43}}\:\right)^{\mathrm{3}/\mathrm{2}} −\left(\mathrm{52}−\mathrm{6}\sqrt{\mathrm{43}}\right)^{\mathrm{3}/\mathrm{2}} }{\mathrm{18}}=? \\ $$ Answered by Rasheed.Sindhi last updated on 28/Oct/24 $$\:\:\frac{\left(\mathrm{52}+\mathrm{6}\sqrt{\mathrm{43}}\:\right)^{\mathrm{3}/\mathrm{2}} −\left(\mathrm{52}−\mathrm{6}\sqrt{\mathrm{43}}\right)^{\mathrm{3}/\mathrm{2}} }{\mathrm{18}}=? \\…
Question Number 213003 by MrGaster last updated on 28/Oct/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{arctan}\:{x}\right)^{\mathrm{2}} {dx}. \\ $$$$ \\ $$ Answered by mathmax last updated on…
Question Number 212997 by cherokeesay last updated on 28/Oct/24 Commented by mr W last updated on 28/Oct/24 $${impossible}!\:{the}\:{height}\:{is}\:{unknown}. \\ $$ Commented by cherokeesay last updated…
Question Number 212999 by golsendro last updated on 28/Oct/24 $$\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{non}\:\mathrm{zero}\:\mathrm{integer}\: \\ $$$$\:\mathrm{solution}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\:\:\:\frac{\mathrm{15}}{\mathrm{x}^{\mathrm{2}} \mathrm{y}}\:+\:\frac{\mathrm{3}}{\mathrm{xy}}\:−\:\frac{\mathrm{2}}{\mathrm{x}}\:=\:\mathrm{2}\: \\ $$ Answered by A5T last updated on 28/Oct/24 $$\frac{\mathrm{3}−\mathrm{2}{y}}{{xy}}=\mathrm{2}−\frac{\mathrm{15}}{{x}^{\mathrm{2}}…
Question Number 212992 by CrispyXYZ last updated on 28/Oct/24 $${a}>\mathrm{0}.\:{b}>\mathrm{0}.\:{a}+{b}=\mathrm{2}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{a}^{\sqrt{{a}}} {b}^{\sqrt{{b}}} . \\ $$ Answered by Ghisom last updated on 28/Oct/24 $${b}=\mathrm{2}−{a} \\…
Question Number 212993 by MrGaster last updated on 28/Oct/24 Commented by MrGaster last updated on 28/Oct/24 "Let ΔABC be inscribed in circle O. Point P is inside ΔABC, and the projections of P on the sides BC, CA, and AB are points X, Y, and Z, respectively. The second intersection point of line AP with circle O is D. Point E lies on circle O, and DE is perpendicular to BC. Let I be the midpoint of DE. Line PI intersects BC at F, and point T lies on AF such that TX is parallel to AD. Prove that line YZ bisects line TX." Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 212925 by MrGaster last updated on 27/Oct/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\:\sqrt{{n}^{\mathrm{2}} +\mathrm{1}}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 212926 by efronzo1 last updated on 27/Oct/24 $$\:\mathrm{find}\:\mathrm{integers}\:\mathrm{x},\mathrm{y}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\frac{\mathrm{x}}{\mathrm{x}−\mathrm{3}}\:−\frac{\mathrm{4}}{\mathrm{y}^{\mathrm{2}} −\mathrm{45}}\:=\:\frac{\mathrm{1}}{\mathrm{100}} \\ $$ Answered by Frix last updated on 27/Oct/24 $$\Leftrightarrow \\ $$$${x}=\frac{\mathrm{3}\left({y}^{\mathrm{2}}…