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$$ \\ $$In a convex quadrilateral ABCD, diagonals AC and BD intersect at E, while perpendicular bisectors of AB and CD intersect at F, and those of BC and DA intersect at G. Prove: (1) E, F, and G are collinear, (2) AE:EC = BF:FD, and (3) CG:GD = AF:FB.
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$${find} \\ $$$$\int\frac{\mathrm{1}}{{x}}{dx} \\ $$
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$$\:\:\mathrm{prove}\:\mathrm{that}\:\mathrm{p}\left(\mathrm{n}\right)\:\mathrm{is}\:\mathrm{integer}\:\forall\:\mathrm{n}\in\mathbb{Z} \\ $$$$\:\:\:\mathrm{p}\left(\mathrm{n}\right)\:=\:\frac{\mathrm{3n}^{\mathrm{7}} +\mathrm{7n}^{\mathrm{3}} +\mathrm{11n}}{\mathrm{21}} \\ $$
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$${given}\:{that}\:{the}\:{roots} \\ $$$$\:{of}\:{the}\:{equation} \\ $$$$\mathrm{3}{x}^{\mathrm{2}} −\left(\mathrm{4}+\mathrm{2}{k}\right){x}+\mathrm{2}{k}=\mathrm{0} \\ $$$${are}\:\alpha\:{and}\:\beta \\ $$$${find}\:{the}\:{value}\:{of}\:{k} \\ $$$${for}\:{which}\:\beta=\mathrm{3}\alpha \\ $$
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$$ \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:{Find}\:\:{the}\:\:{value}\:{of}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} \:\int_{\sqrt{\:{x}^{\:\mathrm{2}} \:+{y}^{\:\mathrm{2}} }} ^{\:\sqrt{\mathrm{2}−{x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} }} \:{xy}\:{dz}\:{dy}\:{dx}\:=?\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\: \\ $$
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