¶ 2022 AMC8 Problem 6
Problem:
Three positive integers are equally spaced on a number line. The middle number is and the largest number is times the smallest number. What is the smallest of these three numbers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let and represent the smallest and largest integers, respectively, with the number between them. The three integers are equally spaced, so and must be equidistant from . Because is greater than and is a multiple of , the smallest possible value of is and hence the smallest possible value of is . If and , the number a will be farther from than . The same is true if and . If and , however, both and will be units from . For larger values of , the number will be closer to than . Therefore and is the only solution, and is the smallest of the three numbers.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions