¶ 2020 AMC8 Problem 22
Problem:
When a positive integer is fed into a machine, the output is a number calculated according to the rule shown below:
For example, starting with an input of , the machine will output . Then if the output is repeatedly inserted into the machine five more times, the final output is .
When the same -step process is applied to a different starting value of , the final output is . What is the sum of all such integers
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Working backwards from the number , as shown below, there are possible starting values of , and . The sum of these values is .
: The Collatz Conjecture states that the number will appear eventually no matter which positive integer is chosen as the starting value. For example if the sequence is extended past , the outputs will be
It is not known whether this conjecture is true.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions