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Question Number 108594 by Rasikh last updated on 18/Aug/20

Answered by 1549442205PVT last updated on 18/Aug/20

$$\mathrm{From}\mathrm{a},\mathrm{b},\mathrm{c}>0\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}=1\mathrm{we}\mathrm{have}$$$$\mathrm{ab}+\mathrm{bc}+\mathrm{ca}=\mathrm{abc}\left(1\right).\mathrm{Apply}{\mathrm{Cauchy}}^{\prime }\mathrm{s}$$$$\mathrm{inequality}\mathrm{for}\mathrm{three}\mathrm{positive}\mathrm{numbers}$$$$\mathrm{we}\mathrm{have}:$$$$\mathrm{abc}=\mathrm{ab}+\mathrm{bc}+\mathrm{ca}⩾3{}^3\sqrt{\mathrm{ab}.\mathrm{bc}.\mathrm{ca}}=3{\left({}^3\sqrt{\mathrm{abc}}\right)}^2$$$$\Rightarrow {}^3\sqrt{\mathrm{abc}}⩾3\left(1\right).\mathrm{Hence},\mathrm{from}\left(1\right)\mathrm{we}\mathrm{have}$$$$\mathrm{P}=(\mathrm{a}-1)(\mathrm{b}-1)(\mathrm{c}-1)=\mathrm{abc}-(\mathrm{ab}+\mathrm{bc}+\mathrm{ca})$$$$+(\mathrm{a}+\mathrm{b}+\mathrm{c})-1=\mathrm{a}+\mathrm{b}+\mathrm{c}-1$$$$⩾3{}^3\sqrt{\mathrm{abc}}-1⩾3.3-1=8$$$$\mathrm{The}\mathrm{equality}\mathrm{occurs}\mathrm{if}\mathrm{an}\mathrm{only}\mathrm{if}$$$$\mathrm{a}=\mathrm{b}=\mathrm{c}=3$$$$\mathrm{Thus},\mathrm{P}=(\mathrm{a}-1)(\mathrm{b}-1)(\mathrm{c}-1)\mathrm{then}$$$${\mathbf{P}}_{\mathbf{min}}=8\mathbf{when}(\mathbf{a},\mathbf{b},\mathbf{c})=(3,3,3)$$

Commented by Rasikh last updated on 18/Aug/20

$$\mathrm{thank}\mathrm{you}$$

Answered by mathmax by abdo last updated on 18/Aug/20

$$\mathrm{we}\mathrm{have}\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}=1\Rightarrow \frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{a}}{\mathrm{c}}=\mathrm{a}-1\mathrm{and}\frac{\mathrm{b}}{\mathrm{a}}+\frac{\mathrm{b}}{\mathrm{c}}=\mathrm{b}-1$$$$\mathrm{and}\frac{\mathrm{c}}{\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{b}}=\mathrm{c}-1\Rightarrow (\mathrm{a}-1)(\mathrm{b}-1)(\mathrm{c}-1)=(\frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{a}}{\mathrm{c}})(\frac{\mathrm{b}}{\mathrm{a}}+\frac{\mathrm{b}}{\mathrm{c}})(\frac{\mathrm{c}}{\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{b}})$$$$⩾2\sqrt{\frac{{\mathrm{a}}^2}{\mathrm{bc}}}.2\sqrt{\frac{{\mathrm{b}}^2}{\mathrm{ac}}}.2\sqrt{\frac{{\mathrm{c}}^2}{\mathrm{ab}}}=8\Rightarrow (\mathrm{a}-1)(\mathrm{b}-1)(\mathrm{c}-1)⩾8\Rightarrow$$$$\min (...)=8$$

Commented by Rasikh last updated on 18/Aug/20

$$\mathrm{thank}\mathrm{you}$$

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