If-a-b-c-5-1-a-1-b-1-c-1-5-what-is-the-greatest-value-of-a-3-b-3-c-3-

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

If {\begin{cases} a + b + c = 5 \\ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5} \end{cases}

what is the greatest value of

a^{3} + b^{3} + c^{3} ?

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

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5} \Rightarrow \frac{ab + bc + ca}{abc} = \frac{1}{5}

a^{3} + b^{3} + c^{3} - 3 abc = (a + b + c) ({(a + b + c)}^{2} - 3 ab - 3 bc - 3 ca)

a^{3} + b^{3} + c^{3} = 5 (5^{2} - 3 (\frac{abc}{5})) + 3 abc

= 5^{3} - 3 abc + 3 abc = 125