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a-b-c-gt-0-Find-the-min-value-of-cyc-a-b-a-b-c-

Question Number 196049 by CrispyXYZ last updated on 17/Aug/23 $${a},\:{b},\:{c}\:>\:\mathrm{0}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{min}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{\mathrm{cyc}} {\sum}\:\sqrt{\frac{{a}+{b}}{{a}+{b}+{c}}}\:. \\ $$ Commented by Frix last updated on 17/Aug/23 $${b}={pa}\wedge{c}={qa} \\ $$$$\Sigma=\frac{\sqrt{{p}+\mathrm{1}}+\sqrt{{p}+{q}}+\sqrt{{q}+\mathrm{1}}}{\:\sqrt{{p}+{q}+\mathrm{1}}}…

Question-195996

Question Number 195996 by sonukgindia last updated on 15/Aug/23 Answered by Rasheed.Sindhi last updated on 15/Aug/23 $${Let}\:{radius}\:{of}\:{blue}\:{quarter}={r} \\ $$$${and}\:{of}\:{green}={R} \\ $$$${R}={r}+\mathrm{1} \\ $$$${Diagonal}={R}+{r} \\ $$$$\left({R}+{r}\right)^{\mathrm{2}}…

Question-196007

Question Number 196007 by sonukgindia last updated on 15/Aug/23 Answered by mr W last updated on 15/Aug/23 $${AC}=\frac{{R}}{\mathrm{tan}\:\frac{\alpha}{\mathrm{2}}} \\ $$$$\overset{\frown} {{BC}}=\left(\pi−\alpha\right){R} \\ $$$$\left(\pi−\alpha\right){R}=\frac{{R}}{\mathrm{tan}\:\frac{\alpha}{\mathrm{2}}} \\ $$$$\left(\pi−\alpha\right)\:\mathrm{tan}\:\frac{\alpha}{\mathrm{2}}=\mathrm{1}…

Question-195954

Question Number 195954 by sonukgindia last updated on 13/Aug/23 Answered by mahdipoor last updated on 13/Aug/23 $$\frac{\mathrm{1}}{\mathrm{4}}×\frac{\left(\mathrm{4}−{a}\right)−{x}}{{x}+\left({a}−\mathrm{2}\right)}×\frac{\left(\mathrm{4}−{a}\right)+{x}}{−{x}+\left({a}−\mathrm{2}\right)}={c} \\ $$$${c}=\frac{\mathrm{1}}{\mathrm{4}}\:\Rightarrow\:\begin{cases}{\mathrm{4}−{a}=\mathrm{2}−{a}\Rightarrow\nexists}\\{\mathrm{4}−{a}={a}−\mathrm{2}\Rightarrow{a}=\mathrm{3}}\end{cases} \\ $$ Answered by Rasheed.Sindhi last…