Draw 4 distinct lines in the plane and let denote the number of intersections (if more than one line passes through the same point it still only counts as one intersection). Which of the following is the set of all possible values of ?
, 2016
Pick one
Solution
As shown in the figures, four distinct lines can intersect in , , , , or points.
Assume that there are exactly two intersections. At each of these either two or three lines meet. Suppose that at one of the intersections, denote it by , three of the lines meet. One of these three lines, call it , has to pass through the other intersection, denoted . The remaining two lines through are denoted by and , and the final line, which also passes through , is denoted by . Now, the line intersects at least one of and in a third intersection, which is a contradiction. So, exactly two lines should meet at each of the two intersections. But in this case the third intersection would again have to exist. The correct answer is (B).
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.