2014 UMO Problems/Problem 4
Joel is playing with ordered lists of integers in the following way. He starts out with an ordered list of nonnegative integers. Then, he counts the number of ’s, ’s, ’s, and so on in the list, writing the counts out as a new list. He stops counting when he has counted everything in the previous list. Then he takes the second list and applies the same process to get a third list. He repeats this process indefinitely.
For example, he could start out with the ordered list . He counts three ’s, zero ’s, and one , and then stops counting, so the second list is In the second list, he counts one , one , zero ’s, and one , so the third list is . Then he counts one and three ’s, so the fourth list is . Here are the first few lists he writes down: If instead he started with , he would write down: If Joel starts out with an arbitrary list of nonnegative integers and then continues this process, there are certain lists of length two that he might end up writing an infinite number of times. Find all such pairs .
Answer: All pairs of work.
Proof: First, note that repeats forever: Now suppose we have . Let We get Otherwise, . If then If we get Otherwise we get Our final case is if . Then we have which we know repeats from .
Therefore, for all it will repeat forever.
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