Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.3 AIME, harder Find the answer

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 $N$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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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