8. On a circle, 2017 different points $A_{1}, \ldots, A_{2017}$ are marked, and all possible chords connecting these points pairwise are drawn. A line is drawn through the point $A_{1}$, not passing through any of the points $A_{2}, \ldots A_{2017}$. Find the maximum possible number of chords that can have at least one common point with this line.

ANSWER: 1018080.
Solution: Let $k$ points be located on one side of the given line, then $2016-k$ points are on the other side. Thus, $k(2016-k)$ chords intersect the line and another 2016 pass through point $A_{1}$. Note that $k(2016-k)=-(k-1008)^{2}+1008^{2}$. The maximum of this expression is achieved when $k=1008$.