[Math] Asian Pacific Mathematics Olympiad (APMO) 2015 Problem 4


Given n blue lines and n red lines in the plane such that no two lines are parallel to each other, show that there is a circle that intersects with the set of blue lines in exactly 2n − 1 points, and intersects with the set of red lines in exactly 2n − 1 points.

Proof:

We will prove a more specific condition, that there exists a circle such that for each color, is tangential to one line and intersects with the rest at two points each. The intuition is to find a red and blue line that "bundle" the rest of 2n − 2 lines, then a circle big enough and tangential to both this red and blue line indeed intersects the rest of 2n − 2 lines at two points each.

To find such two lines, it suffices to take the blue and red lines that span the largest angle between them. Even these two lines may not be unique, it is not hard to show that a sufficiently big circle tangential to them intersects each of other lines at two points.

Q.E.D.


post by Shen-Fu Tsai


References:

[1]Asian Pacific Mathematics Olympiad