Problem:
Suppose a,b, and c are nonzero real numbers, and a+b+c=0. What are the possible value(s) for ∣a∣a+∣b∣b+∣c∣c+∣abc∣abc?
Answer Choices:
A. 0
B. 1 and −1
C. 2 and −2
D. 0,2, and −2
E. 0,1, and −1
Solution:
Since a+b+c=0, then these three numbers cannot be all positive or all negative. The value of ∣X∣X=1 for X positive and −1 for X negative.
Case I. When there are two positive numbers and one negative number,
∣a∣a+∣b∣b+∣c∣c=1,
and ∣abc∣abc=−1, so
∣a∣a+∣b∣b+∣c∣c+∣abc∣abc=0.
Case II. When there are two negative numbers and one positive number,
∣a∣a+∣b∣b+∣c∣c=−1,
and ∣abc∣abc=1, so
∣a∣a+∣b∣b+∣c∣c+∣abc∣abc=0.
Therefore the only possible value of ∣a∣a+∣b∣b+∣c∣c+∣abc∣abc is 0.
Answer: A.
The problems on this page are the property of the MAA's American Mathematics Competitions