Let S={1,2,…,1010}. Find all functions f:S→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)).
Source: Serbia TST 2021, P6
Tags: number theory, algebra
Let S={1,2,…,1010}. Find all functions f:S→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)).