Team Selection Test for IMO 2023

Given a good sequence, consider the following for all $n$ for fixed $i$, $j$, $k$: $$n\mid a_{i+k}-a_i\iff f(n)\mid k\iff n\mid a_{j+k}-a_j.$$ Thus one has $$|a_{i+k}-a_i|=|a_{j+k}-a_j|$$ for all $i$, $j$, $k$. Now consider two cases separately:

a. If $a_1=a_3$, then $a_{i+2}=a_i$ for all $i$, which means $$a_i=\{\begin{array}{ll}{c}_0,& i\equiv 0(\mathrm{mod}2),\\ c_1,& i\equiv 1(\mathrm{mod}2)\end{array}$$ for some $c_0$, $c_1$ which are not necessarily distinct. Indeed, then the function $$f(n)=\{\begin{array}{ll}1,& n\mid c_0-c_1,\\ 2,& n\nmid c_0-c_1\end{array}$$ satisfies the condition.

b. If $a_1\ne a_3$, then $a_{i+2}\ne a_i$ for all $i$, thus $(a_{i+2}-a_{i+1})\ne -(a_{i+1}-a_i)$, hence $(a_{i+2}-a_{i+1})=(a_{i+1}-a_i)$, which means $$a_i=ki+c$$ for some $k$ and $c$. Indeed, then the function $$f(n)=\frac{n}{\gcd (n,k)}$$ satisfies the condition.