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HMMT Algebra/NT 2019/4: Binary recursive function

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February 17, 2019
HMMTalgebrafunction

Problem Statement

Let N\mathbb{N} be the set of positive integers, and let f:N→Nf: \mathbb{N} \to \mathbb{N} be a function satisfying
[*] f(1)=1f(1) = 1, [*] for n∈Nn \in \mathbb{N}, f(2n)=2f(n)f(2n) = 2f(n) and f(2n+1)=2f(n)−1f(2n+1) = 2f(n) - 1.
Determine the sum of all positive integer solutions to f(x)=19f(x) = 19 that do not exceed 2019.
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