2003 AMC 12B Problem 24

Positive integers a,b, and c are chosen so that a<b<c, and the system of equations

2x + y = 2003 \quad and \quad y = |x-a| + |x-b| + |x-c|

has exactly one solution. What is the minimum value of c?

\mathrm{(A)}\ 668 \qquad\mathrm{(B)}\ 669 \qquad\mathrm{(C)}\ 1002 \qquad\mathrm{(D)}\ 2003 \qquad\mathrm{(E)}\ 2004

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.