¶ 2022 AMC8 Problem 13
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Problem:
How many positive integers can fill the blank in the sentence below? “One positive integer is more than twice another, and the sum of the two numbers is .”
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be the missing number, and let and be the two numbers that sum to , with . Because is more than twice , the value of cannot exceed of . It follows that the possible values of are the integers from to . The table below shows the corresponding values of , which equals the difference between and , and the values of , which equals the difference between and
Each value of produces a different value of , so there are values of that can fill the blank in the given sentence.
Let be the missing number, and let a be the smaller of the two numbers in the problem. Then the larger number is , and their sum is
Both and are positive integers so the possible values of are the integers from to , each corresponding to a distinct positive integer . Thus there are values of that can fill the blank in the given sentence
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions