Correct solutions were also received from Jan Siwanowicz and from George Kirkup. Two incorrect solutions were submitted.
As two solvers pointed out, the condition on the non-colinearity of any set of three points is unnecessary. The following solution was submitted by Jan Siwanowicz.
Let S denote the set of slopes of lines joining two of the 1998 points. This is a finite set so there is some real number m not in the set. For each of the 1998 points the line of slope m passing through that point intercepts the y-axis at some point, let y1,..., y1998 be those points, labeled in such a way as to have y1 > y2 > ... > y1998. Notice that all of these points are different. Pick any point in the interval strictly between y999 and y1000 and draw the line through this point with slope m. That'll do it!
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