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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 .
does not have a real root. what is the maximum number of coefficients to be 
?(14 points)
needs have no real roots and 
, it means we need have 
for all real 
. Since we then need 
, it means the number of negative coefficients can be at most 
(out of the total 
coefficients). negative coefficients. Take 
. But 
, while 
. But a polynomial 
has only one real root (root 
), so polynomial 
has no real roots, hence is positively defined. Therefore all terms in the sum above are non-negative, and may be zero only at values 
and/or . However, 
, so 
has no real roots.
has only one real root (root ), so polynomial 
has no real roots.
does not have a real root. what is the maximum number of coefficients to be ?(14 points)
, we see that there doesn't may more than 
, each term is positive.
and 
and 
and add it, we find 
:
and , add them and we find 
in natural numbers m and n.
