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A strange distcrete functional equation seems analytic

Source: kjmo 2022 P4

October 29, 2022

algebrafunctional equationfunction

Problem Statement

Find all function

f:\mathbb{N} \longrightarrow \mathbb{N}

such that
forall positive integers

x

and

y

\frac{f(x+y)-f(x)}{f(y)}

is again a positive integer not exceeding

2022^{2022}