Problem:
Two particles move along the edges of equilateral in the direction
starting simultaneously and moving at the same speed. One starts at , and the other starts at the midpoint of . The midpoint of the line segment joining the two particles traces out a path that encloses a region . What is the ratio of the area of to the area of ?
Answer Choices:
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B.
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E.
Solution:
Imagine a third particle that moves in such a way that it is always halfway between the first two. Let , and denote the midpoints of , and , respectively, and let , and denote the midpoints of , and , respectively. When the first particle is at , the second is at and the third is at . When the first particle is at , the second is at and the third is at . Between those two instants, both coordinates of the first two particles are linear functions of time. Because the average of two linear functions is linear, the third particle traverses . Similarly, the third particle traverses as the first traverses and the second traverses . Finally, as the first particle traverses and the second traverses , the third traverses . As the first two particles return to and , respectively, the third makes a second circuit of .
If is the center of , then by symmetry is also the center of equilateral . Note that
so the ratio of the area of to that of is
The problems on this page are the property of the MAA's American Mathematics Competitions