Problem:
Let be the set of points in the coordinate plane such that two of the three quantities , and are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of ?
Answer Choices:
A. a single point
B. two intersecting lines
C. three lines whose pairwise intersections are three distinct points
D. a triangle
E. three rays with a common endpoint
Solution:
Suppose that the two larger quantities are the first and the second. Then . This is equivalent to and , and its graph is the downward-pointing ray with endpoint . Similarly, if the two larger quantities are the first and third, then . This is equivalent to and , and its graph is the leftward-pointing ray with endpoint . Finally, if the two larger quantities are the second and third, then . This is equivalent to and , and its graph is the ray with endpoint that points upward and to the right. Thus the graph consists of .
Note: This problem is related to a relatively new area of mathematics called tropical geometry.
The problems on this page are the property of the MAA's American Mathematics Competitions