Question 27
S5 (FIN). For positive integers $n$, the numbers $f(n)$ are defined inductively as follows: $f(1)=1$, and for every positive integer $n$, $f(n+1)$ is the greatest integer $m$ such that there is an arithmetic progression of positive integers $a_1\lt a_2\lt \cdots\lt a_m=n$ for which
$$f(a_1)=f(a_2)=\cdots=f(a_m).$$
Prove that there are positive integers $a$ and $b$ such that $f(an+b)=n+2$ for every positive integer $n$.
$$f(a_1)=f(a_2)=\cdots=f(a_m).$$
Prove that there are positive integers $a$ and $b$ such that $f(an+b)=n+2$ for every positive integer $n$.