1. Answer: $2011^{2011}+2009^{2009}>2011^{2009}+2009^{2011}$.
Solution. Transform the difference between the left and right parts:

$$
\begin{aligned}
& 2011^{2011}+2009^{2009}-\left(2011^{2009}+2009^{2011}\right)= \\
& =2011^{2011}-2011^{2009}-\left(2009^{2011}-2009^{2009}\right)= \\
& =2011^{2009}\left(2011^{2}-1\right)-2009^{2009}\left(2009^{2}-1\right)
\end{aligned}
$$

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