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Question Number 106288 by ZiYangLee last updated on 04/Aug/20

$$\mathrm{If}0<\mathrm{x}<1<\mathrm{y}<100<\mathrm{z},\mathrm{and}\mathrm{satisfy}$$ $$\mathrm{these}\mathrm{equations}:$$ $$\{\begin{array}{l}\log {}_2\left(\mathrm{xyz}\right)=103\\ \frac{1}{{\log }_2\mathrm{x}}+\frac{1}{{\log }_2\mathrm{y}}+\frac{1}{{\log }_2\mathrm{z}}=\frac{1}{103}\end{array}$$ $$\mathrm{Find}\mathrm{xyz}(\mathrm{x}+\mathrm{y}+\mathrm{z})-\mathrm{xy}-\mathrm{yz}-\mathrm{zx}$$

Answered by bemath last updated on 04/Aug/20

$$\left(1\right)\mathrm{xyz}=2^{103}$$ $$\left(2\right)\frac{1}{\log {}_2\mathrm{x}}+\frac{1}{\log {}_2\left(\frac{2^{103}}{\mathrm{xz}}\right)}+\frac{1}{\log {}_2\left(\mathrm{z}\right)}=\frac{1}{103}$$ $$\frac{1}{\log {}_2\mathrm{x}}+\frac{1}{103-(\log {}_2\mathrm{x}+\log {}_2\mathrm{z})}+\frac{1}{\log {}_2\mathrm{z}}=\frac{1}{103}$$ $$$$

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