Nemotron-IMO-Bench 2026 N1

Let $(a_n)_{n\geq 1}$ be a sequence of positive integers such that $a_1=1$ and for any prime $p$ the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.