Problem

Source: Serbia TST 2021, P6

Tags: number theory, algebra



Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$for each $x \in S$ (assume $f(10^{10}+1)=f(1)$).