Competition problem — Geometry — Maths Olympiad Prep

Library / /1020 of 1394

Geometry Difficulty 5.5 AIME, harder Prove it United States

Problem:

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 $N$ .

Solution

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 $6+2+2+1=11$ .

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