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Question Number 140275 by liberty last updated on 06/May/21

$$\mathrm{If}\{\begin{array}{l}\mathrm{a}+\mathrm{b}+\mathrm{c}=5\\ \frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}=\frac{1}{5}\end{array}$$$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{greatest}\mathrm{value}\mathrm{of}$$$${\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3?$$

Answered by Ar Brandon last updated on 06/May/21

$$\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}=\frac{1}{5}\Rightarrow \frac{\mathrm{ab}+\mathrm{bc}+\mathrm{ca}}{\mathrm{abc}}=\frac{1}{5}$$$${\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3-3\mathrm{abc}=(\mathrm{a}+\mathrm{b}+\mathrm{c})({(\mathrm{a}+\mathrm{b}+\mathrm{c})}^2-3\mathrm{ab}-3\mathrm{bc}-3\mathrm{ca})$$$${\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=5(5^2-3\left(\frac{\mathrm{abc}}{5}\right))+3\mathrm{abc}$$$$=5^3-3\mathrm{abc}+3\mathrm{abc}=125$$

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