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