EGMO 2025 Problem 2

An infinite increasing sequence $a_{1}< a_{2}< a_{3}< \dots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_{n}$ terms of the sequence is equal to $a_{n}$ .

Show that there exists an infinite sequence $b_{1},b_{2},b_{3},\ldots$ of positive integers such that for every central sequence $a_{1},a_{2},a_{3},\ldots$ , there are infinitely many positive integers $n$ with $a_{n} = b_{n}$ .