Problem:
For how many integer values of do the graphs of and not intersect?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
When , the graphs of and consist of the single point and the union of the two lines and , respectively; so the two graphs intersect. When , the graph of is a circle of radius centered at the origin and the graph of is an equilateral hyperbola centered at the origin. The vertices of the hyperbola, located at\
if or at if , are the closest points on the graph to the origin. If , then
thus the graphs intersect. If , then
and thus the graphs do not intersect. Thus the graphs do not intersect for or .
The problems on this page are the property of the MAA's American Mathematics Competitions