<s>[INST] Solve $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$ Solve for $x$
$\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$  

solution should be by hand

 [/INST] First it is clear that $x \geq 0$.  
By AM-GM inequality we have:
$$x^{2010}+1 \geq 2 x^{1005}$$
$$1+x^2+x^4+x^6+.......+x^{2008} \geq 1005 \sqrt[1005]{1x^2...x^{2008}}=1005 x^{\frac{2+4+6+...+2008}{1005}}=1005x^{1004}$$
Multiplying you get
$$\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right) \geq2010x^{2009}$$
You get equality if and only if you have equality in the above. Thus
$$x^{2010} =1$$
and
$$1=x^2=x^4=...=x^{2008}$$
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