which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
are given such that 
.
in natural numbers m and n.
,
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
Y by Adventure10, Mango247, and 1 other user
Prove that the number of pairs 
of a permutation 
of 
and a subset 
of such that
![\[ \forall x\in S: \alpha(x)\not\in S\]](//latex.artofproblemsolving.com/3/b/5/3b5b87de58ac757a2eb09ddeb7989dcbd6b8270d.png)
is equal to
in which 
is the Fibonacci sequence such that 
of a permutation 
of 
and a subset 
of such that![\[ \forall x\in S: \alpha(x)\not\in S\]](http://latex.artofproblemsolving.com/3/b/5/3b5b87de58ac757a2eb09ddeb7989dcbd6b8270d.png)
is equal to
in which 
is the Fibonacci sequence such that 
elements we should pick these 
place and fill other blanks with other elements so the nimber is:
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