Problem:
Let , and be real numbers satisfying the system of equations
Let be the set of possible values of . Find the sum of the squares of the elements of .
Solution:
Answer (273):
First, consider the case when . Then , implying that or . In both cases, all three equations are satisfied.
If not all of are equal, without loss of generality let be different from both and . Subtracting the second equation from the first yields
from which . Similarly, , implying that . Thus if not all of are equal, one of the variables must be equal to 11 while the other two are equal to 4 , in which case all three equations are satisfied.
Hence the set of possible values of is . The requested sum of squares is .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions