Question 36
Let $a$, $b$ be positive integers and let $f: \mathbb{N} \to \mathbb{N}$ be a function such that for each positive integer $n$, the number $f(n+a)$ is divisible by $f([\sqrt{n}] + b)$. Prove that for each positive integer $n$ there exist $n$ pairwise distinct and pairwise relatively prime positive integers $a_1, a_2, \dots, a_n$ such that the number $f(a_{i+1})$ is divisible by $f(a_i)$ for all $i = 1, 2, \dots, n-1$. (Here $[x]$ is the integer part of $x$, i.e. the greatest integer not exceeding $x$; $\mathbb{N}$ is the set of positive integers.)