Problem:
Define x♡y to be ∣x−y∣ for all real numbers x and y. Which of the following statements is not true?
Answer Choices:
A. x♡y=y♡x for all x and y
B. 2(x♡y)=(2x)♡(2y) for all x and y
C. x♡0=x for all x
D. x♡x=0 for all x
E. x♡y>0 if xî€ =y
Solution:
For example, −1♡0=∣−1−0∣=1î€ =−1. All the other statements are true:
(A) x♡y=∣x−y∣=∣−(y−x)∣=∣y−x∣=y♡x for all x and y.
(B) 2(x♡y)=2∣x−y∣=∣2x−2y∣=(2x)♡(2y) for all x and y.
(D) x♡x=∣x−x∣=0 for all x.
(E) x♡y=∣x−y∣>0 if xî€ =y.
Answer: C​.
The problems on this page are the property of the MAA's American Mathematics Competitions