10. Let $[x]$ denote the greatest integer not exceeding $x$, then the number of elements in the set $\left\{\left[\frac{1^{2}}{2005}\right],\left[\frac{2^{2}}{2005}\right], \ldots,\left[\frac{2005^{2}}{2005}\right]\right\}$ is . $\qquad$
10. 1501 When $\frac{(n+1)^{2}}{2005}-\frac{n^{2}}{2005} \geqslant 1$ i.e., $n \geqslant 1002$, $\left[\frac{(n+1)^{2}}{2005}\right]$ and $\left[\frac{n^{2}}{2005}\right]$ are not the same, so the number of elements in the set for $2005 \geqslant n \geqslant 1002$ is 1004;

When $n \leqslant 1001$, $\left[\frac{(n+1)^{2}}{2005}\right]-\left[\frac{n^{2}}{2005}\right]=0$ or 1, i.e., $\left[\frac{k^{2}}{2005}\right](k=1,2, \cdots, 1001)$ takes all integers from $0,1,2, \cdots$, $\left[\frac{1001^{2}}{2005}\right]$, so the number of elements in the set for $n \leqslant 1001$ is $\left[\frac{1001^{2}}{2005}\right]+1=497$.
In summary, the original set contains $1004+497=1501$ elements.