A problem on Tournament of the towns 2002 Fall/A-level:
A sequence with first two terms equal to $1$ and $24$ respectively, is defined by the following rule: each subsequent term is equal to the smallest positive integer which has not yet occurred in the sequence and is not coprime with the previous term. Prove that all positive integers occur in this sequence.
Let $a_k$ be the k-th element in the sequence. Proving that for all $n$, there exists a $k = k(n)$, such that $a_k = n$ is a fun exercise. Being able to upper bound $k$ is already non-trivial. And proving that $n ~ \sim k(n)$ -which I tried for quite some time, but failed- unless $n$ is a prime or three times a prime is research level. Just a few minutes ago I found out that Hofman and Pilipczuk proved it three years ago; http://www.mimuw.edu.pl/~malcin/ekg.pdf
