6. The graph of the quadratic function $y=-\frac{1}{2} x^{2}+\frac{1999}{2} x+1000$ passes through ( ) integer lattice points in the first quadrant (i.e., points with positive integer coordinates).
(A) 1000
(B) 1001
(C) 1999
(D) 2001
6. (C).
$y=-\frac{1}{2}(x-2000)(x+1)$ The range of the independent variable in the first quadrant is $0<x<2000$. When $x$ is odd, $x+1$ is even; when $x$ is even, $x-2000$ is even. Therefore, when $x=1,2, \cdots, 1999$, $y$ takes integer values.
Thus, the number of lattice points in the first quadrant is 1999.