Problem

Source: 2023 OLCOMA Costa Rica National Olympiad, Final Round, 3.1

Tags: Functional Equations, algebra, nonnegative integers



Let $\mathbb Z^{\geq 0}$ be the set of all non-negative integers. Consider a function $f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}$ such that $f(0)=1$ and $f(1)=1$, and that for any integer $n \geq 1$, we have \[f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\]Determine the value of $f(2023)/f(2022)$.