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n-> n/2 if n is even, n -> n62+3 if n is odd

Source: Netherlands - Dutch NMO 2023 p3

March 25, 2024
number theorycombinatorics

Problem Statement

Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the next step: he replaces the number nn on the board by 12n\frac12n if nn is even and by n2+3n^2 + 3 if nn is odd. For how many choices of starting numbers below 20232023 will Felix never write a number of more than four digits on the board?
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