Problem:
Define x⋄y to be ∣x−y∣ for all real numbers x and y. What is the value of
(1⋄(2⋄3))−((1⋄2)⋄3) ?
Answer Choices:
A. −2
B. −1
C. 0
D. 1
E. 2
Solution:
First observe that 2⋄3=∣2−3∣=∣−1∣=1 and 1⋄2=∣1−2∣=∣−1∣=1. Therefore
(1⋄(2⋄3))−((1⋄2)⋄3)=1⋄1−1⋄3=∣1−1∣−∣1−3∣=∣0∣−∣−2∣=(A)−2.
Note: This shows that the ⋄ operation is not associative.
The problems on this page are the property of the MAA's American Mathematics Competitions