Problem:
In a given plane, points and are units apart. How many points are there in the plane such that the perimeter of is units and the area of is square units?
Answer Choices:
A.
B.
C.
D.
E. infinitely many
Solution:
In order for the area of to be , the altitude to the base must be . Thus must lie on one of the two lines parallel to and units from line . In order for the perimeter of to be , the sum of the lengths of the other two sides must be , which implies that point lies on an ellipse whose foci are and and whose semi-minor axis has length , which is less than . Therefore the ellipse does not intersect either of the parallel lines, and there are no such points .
As above, the altitude to the base of length is . Therefore and , and at least one of those sides has length greater than . This contradicts the fact that the perimeter is , so no such points exist.
The problems on this page are the property of the MAA's American Mathematics Competitions