ELMO SL 2010 C2

For a positive integer $n$ , let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$ . Show that for every real number $x$ , there exists an integer $N$ such that for all $n > N$ , $$\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.$$

Alex Zhu.