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Question Number 142639 by ajfour last updated on 03/Jun/21

Commented by ajfour last updated on 03/Jun/21

$$Findmaximumvalueof$$$$\left(AC\right)\left(CD\right)\left(DB\right).$$

Commented by mr W last updated on 04/Jun/21

$$nicequestion!$$$$iguessitshouldbeAC=CD=DB$$

Answered by 1549442205PVT last updated on 04/Jun/21

Answered by 1549442205PVT last updated on 05/Jun/21

$$\mathrm{Let}\mathrm{O},\mathrm{P}\mathrm{be}\mathrm{the}\mathrm{centers}\mathrm{of}\mathrm{two}\mathrm{circles}$$$$\mathrm{Denote}\mathrm{by}\mathrm{K}\mathrm{the}\mathrm{projection}\mathrm{of}\mathrm{O}\mathrm{on}\mathrm{the}$$$$\mathrm{PF}\Rightarrow 2^2={\mathrm{OP}}^2={\mathrm{OK}}^2+{(\mathrm{OF}-\mathrm{OE})}^2$$$$\Rightarrow {\mathrm{EF}}^2={\mathrm{OK}}^2=4-{(\mathrm{OF}-\mathrm{OE})}^2⩽4$$$$\Rightarrow \mathrm{CD}+\frac{\mathrm{AC}+\mathrm{DB}}{2}⩽2\Rightarrow 2\mathrm{CD}+\mathrm{AC}+\mathrm{DB}⩽4$$$$\Rightarrow 2\mathrm{CD}.\mathrm{AC}.\mathrm{DB}⩽{\left(\frac{2\mathrm{CD}+\mathrm{AC}+\mathrm{BD}}{3}\right)}^3⩽\frac{64}{27}$$$$\Rightarrow \mathrm{CD}.\mathrm{AC}.\mathrm{DB}⩽\frac{32}{27}.\mathrm{Hence}$$$${\mathrm{AC}.\mathrm{CD}.\mathrm{DB})}_{\max }=\frac{32}{27}$$$$\mathrm{when}\mathrm{CD}=\frac{\mathrm{AC}}{2}=\frac{\mathrm{BD}}{2}=\frac{2}{3},\mathrm{OE}=\mathrm{PF}=\frac{\sqrt{5}}{3}$$

Commented by mr W last updated on 04/Jun/21

$$nicesolutionsir!$$

Commented by ajfour last updated on 05/Jun/21

$$Indded,thanksSir.$$

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