ELMO 2025 Problem 5

Let $k$ be a positive integer. Define a sequence of positive integers $a_1$ , $a_2$ , $\dots$ by $a_1=1$ and $$a_{n+1}=a_n+k^{a_n}$$for all positive integers $n$ . Show that there exist infinitely many primes $p$ such that for any integer $r$ , there exists an index $m$ such that $a_m\equiv r\pmod p$ .

Oron Wang