HMMT 二月 2022 · 冲刺赛 · 第 9 题

HMMT February 2022 — Guts Round — Problem 9

[7] An E -shape is a geometric figure in the two-dimensional plane consisting of three rays pointing in the same direction, along with a line segment such that • the endpoints of the rays all lie on the segment, • the segment is perpendicular to all three rays, • both endpoints of the segment are endpoints of rays. Suppose two E -shapes intersect each other N times in the plane for some positive integer N . Compute the maximum possible value of N .

解析

[7] An E -shape is a geometric figure in the two-dimensional plane consisting of three rays pointing in the same direction, along with a line segment such that • the endpoints of the rays all lie on the segment, • the segment is perpendicular to all three rays, • both endpoints of the segment are endpoints of rays. Suppose two E -shapes intersect each other N times in the plane for some positive integer N . Compute the maximum possible value of N . Proposed by: Daniel Xianzhe Hong Answer: 11 Solution: Define a C -shape to be an E -shape without the middle ray. Then, an E -shape consists of a ray and a C -shape. Two C -shapes can intersect at most 6 times, a C -shape and a ray can intersect at most 2 times, and two rays can intersect at most 1 time. Thus, the number of intersections of two E -shapes is at most 6 + 2 + 2 + 1 = 11.