[Numerical inequalities. Comparing numbers.]

What is greater: $2011^{2011}+2009^{2009}$ or $2011^{2009}+2009^{2011}$?
Let's transform the difference between the left and right parts:

$2011^{2011}+2009^{2009}-\left(2011^{2009}+2009^{2011}\right)=2011^{2011}-2011^{2009}-\left(2009^{2011}-2009^{2009}\right)=2011^{2009}\left(2011^{2}-1\right)-$ $2009^{2009}\left(2009^{2}-1\right)$.

Notice that $2011^{2009}>2009^{2009}>0$ and $2011^{2}-1>2009^{2}-1>0$. Therefore, the minuend is greater than the subtrahend, which means the difference is positive.

## Answer

$2011^{2011}+2009^{2009}>2011^{2009}+2009^{2011}$.