A friend of mine told me about a contest problem, which he had once seen but the provenance of which he could not remember:
An infinite sequence of positive integers $a_i$ has this property: for every prime $p$, the set $\{ a_1,a_2,...,a_p\}$ is a complete residue system modulo $p$. Now prove that $\frac{a_n}{n}$ converges to $1$.
1) Does anyone recognise this problem and know whence it comes?
2) I would like a solution. Though I have not solved it myself, I have made a curious observation. Denoting the primes by $p_i$, the sequence with these blocks $[p_n,p_n-1,...,p_{n-1}+1]$ is admissible. The statement in the problem then implies that $\frac{p_{n+1}}{p_n}$ tends to $1$. Rarely does a contest problem say anything about the distribution of the primes.
EDIT: My curiosity about this problem is growing day by day, but my progress is not. I would be thrilled about and grateful for a solution, whatever techniques are employed.
This is problem 4 from the 2015 Miklós Schweitzer contest. There is an AoPS thread discussing this problem, see here. A couple of solutions is given there.