HMMT 十一月 2014 · THM 赛 · 第 6 题

HMMT November 2014 — THM Round — Problem 6

Let P , P , P be pairwise distinct parabolas in the plane. Find the maximum possible number of 1 2 3 intersections between two or more of the P . In other words, find the maximum number of points that i can lie on two or more of the parabolas P , P , P . 1 2 3

解析

Let P , P , P be pairwise distinct parabolas in the plane. Find the maximum possible number of 1 2 3 intersections between two or more of the P . In other words, find the maximum number of points that i can lie on two or more of the parabolas P , P , P . 1 2 3 Answer: 12 Note that two distinct parabolas intersect in at most 4 points, which is not difficult 1 to see by drawing examples. Given three parabolas, each pair intersects in at most 4 points, for at most 4 · 3 = 12 points of intersection in total. It is easy to draw an example achieving this maximum, for example, by slanting the parabolas at different angles.