<s>[INST] Find the value of $(a^3 + b^3 + c^3)/(abc)$ if $a/b + b/c + c/a = 1$. 
Find the value of $$\frac{a^3+b^3+c^3}{abc}\qquad\text{ if }\quad \frac ab + \frac bc + \frac ca = 1.$$

I tried using Cauchy's inequality but it was of no help. Please guide me. 
$a, b, c$ are real.
 [/INST] Let 
$$\begin{eqnarray}
u&=&a+b+c\\
v&=&a^3+b^3+c^3\\
s&=&a^2b + b^2 c + c^2 a\\
t &=& a \, b\, c
\end{eqnarray}$$
These are related by 
$$u^3 = v + 3 s + t $$
Further, we are given
$$ \frac ab + \frac bc + \frac ca = \frac st =1$$
so $$u^3 = v + 4 t$$
And our target is
$$\frac{a^3+b^3+c^3}{abc}=\frac vt = \frac{u^3}{t}-4$$
This cannot be a single value, as we have a remaining degree of freedom. 
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