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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 
. Prove the inequality 
, let 
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
. Determine for each positive integer .
of positive odd integers, such that 
and 
.
such that 
and 
must be positive, and 
is obviously a solution.
must be positive, and is obviously a solution.
and 
. If 
and , then 
and 
- solution.
or 
.
, then 
or 
are roots of 
with 
.
. For 
k=1.
. Note that 
, therefore 
.
.
,or 
.Considering all the cases
.substituting we can find the solutions .
or ,Which can be analyzed as previous.
and 
are even numbers(the later being divisible by 
) since are odd.So the conditions can be extended to
and 
and 
.
and 
gives us the only solution 
.For 
we have 
(not possible) and the other case yeilds 
(not possible).So no solution in this case.
as the only solution.
as the only solution which is not counted before.
.But note that this is not possible as LHS is 
as the only solutions.
we have 
and 
and 
is even)
we have 
note that 
let 
is odd, we also have 
is odd, we also have 
we have 
absurd 
we split the problem into several cases:
are 
is pretty much the same for all 
small cases to check, instead of only 
, namely for even 
and for odd ). Here is a solution to the problem for all 
, .
in natural numbers m and n.
.
;
.
;
- no solution.
;
and 
,
.
,![\[\dfrac{nk-1}{3}\mid n^2+3\Rightarrow nk-1\mid 3n^2k+9k\Rightarrow nk-1\mid 9k+3n\]](http://latex.artofproblemsolving.com/3/c/3/3c3e0b4dc6ceeec813f389bd157b9e3be74526d9.png)
![\[nk-1\leq 9k+3n\Leftrightarrow (n-9)(k-3)\leq 28\]](http://latex.artofproblemsolving.com/e/9/f/e9f5081f4afcccb5abc5b62c54e184e0fb605abb.png)
,
,![\[4n-1\mid 36+3n\Rightarrow 4n-1\mid 147\Rightarrow n\in \left \{ 1,37 \right \}\]](http://latex.artofproblemsolving.com/7/a/8/7a85eafd6784170934aaf9acd39488cd6bb9608a.png)
.![\[n-9\leq \dfrac{28}{k-3}\leq \dfrac{28}{3}\Rightarrow n\leq 18\]](http://latex.artofproblemsolving.com/c/d/5/cd52a8ad68c9ccacd6c8525677350c0cd0f500d2.png)
.![\[(m,n)=(1,1),(43,13),(49,37)\]](http://latex.artofproblemsolving.com/c/0/2/c027931b2dd9f8bf9ccc028556fe26c2afa060af.png)
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
for some positive integer 
.
and we want 
to divide 
,
to divide 
.
,
Hence 
,
.
and 
,
,
and 
.
.
we want 
to divide 
and get 
,
to divide 
,
.
to divide 
,
.
,
divides 
,
only 
gives remainder 
,
.
,
divides 
,
,
.
,
divides 
,
,
divides 
,
,
.
.
we want 
to divide 
.
to divide 
,
.
we want 
to divide 
and the only solution is 
.
we want 
to divide 
,
.
we want 
to divide 
,
.
we want 
to divide 
,
we want 
to divide 
,
(as 
is prime). For 
we want 
to divide 
,
.