Problem:
Under what conditions is true, where and are real numbers?
Answer Choices:
A. It is never true.
B. It is true if and only if .
C. It is true if and only if .
D. It is true if and only if and .
E. It is always true.
Solution:
Squaring both sides produces
which is equivalent to . Thus is a necessary condition for to be true. But if and only if or , in which case the given equation becomes either or , which are true if and only if or , respectively. It follows that if and only if and . In other words, for the given equation to be true, one of the variables must be 0 and the other must be nonnegative.
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions