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Category: Geometry

Question-214536

Question Number 214536 by ajfour last updated on 11/Dec/24 Commented by ajfour last updated on 11/Dec/24 $${Out}\:{of}\:{questions}\:{labelled}\:{by}\:{just} \\ $$$${permutations}\:{of}\:{digits}\:{of}\:{the}\:{number} \\ $$$$\mathrm{123456}\:\:\:{what}\:{rank}\:{does}\:{this}\:{Q}\:{hold}? \\ $$$${In}\:{increasing}\:{order},\:{that}\:{is}. \\ $$…

Question-214350

Question Number 214350 by ajfour last updated on 06/Dec/24 Answered by mr W last updated on 06/Dec/24 $$\mathrm{tan}\:\alpha=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\mathrm{tan}\:\frac{\alpha}{\mathrm{2}}=\sqrt{\mathrm{5}}−\mathrm{2} \\ $$$$\mathrm{2}+{r}\:\mathrm{tan}\:\frac{\alpha}{\mathrm{2}}=\sqrt{\left({r}+\mathrm{1}\right)^{\mathrm{2}} −\left({r}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\mathrm{2}+\left(\sqrt{\mathrm{5}}−\mathrm{2}\right){r}=\mathrm{2}\sqrt{{r}} \\…