51st Ukrainian National Mathematical Olympiad, 3rd Round

$$n^3+n^2+330n+330=(n+1)(n^2+330),$$ So this expression is divisible by $2011$ if at least one bracket is divisible by $2011$. If first bracket is divisible by $2011$ then minimal $n=2010$, for $n^2+330$, since $n^2$ is increasing, we find that for $n=41$, $n^2+330=2011$.