Problem:
Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are , and years. The positions of the star and all three planets are currently collinear. They will next be collinear after years. Find .
Solution:
All four positions will be collinear if and only if the difference in the number of revolutions made by each pair of planets is an integer multiple of . When the outermost planet has made revolutions, the middle and innermost planets will have made and revolutions, respectively. Thus it is necessary and sufficient that for some integer , so the smallest positive solution for is . Hence .
The problems on this page are the property of the MAA's American Mathematics Competitions