5. $[x]$ represents the greatest integer not greater than $x$. In the set $\left\{\left[\frac{1^{2}}{2003}\right],\left[\frac{2^{2}}{2003}\right], \cdots \cdots,\left[\frac{2003^{2}}{2003}\right]\right\}$, the number of distinct elements is ( ).
A. 1503
B. 1504
c. 1505
D. 2003
5. 1

When $\frac{(n+1)^{2}}{2003}-\frac{n^{2}}{2003} \geqslant 1$, i.e., $n \geqslant 1001$, $\left[\frac{(n+1)^{2}}{2003}\right]$ and $\left[\frac{n^{2}}{2003}\right]$ are distinct, so there are 1003 elements for $n \geqslant 1001$; when $\frac{(n+1)^{2}}{2003}-\frac{n^{2}}{2003}<1$, i.e., $n \leqslant 1000$, the difference between two adjacent elements is 1 or 0, so the number of elements is $\left[\frac{1000^{2}}{2003}\right]+1=500$. In summary, the set contains 1503 elements.