EGMO 2022 Problem 3

An infinite sequence of positive integers $a_{1}, a_{2}, \ldots$ is called good if (1) $a_{1}$ is a perfect square, and (2) for any integer $n \geq 2, a_{n}$ is the smallest positive integer such that

$$n a_{1}+(n-1) a_{2}+\ldots+2 a_{n-1}+a_{n}$$

is a perfect square. Prove that for any good sequence $a_{1}, a_{2}, \ldots$, there exists a positive integer $k$ such that $a_{n}=a_{k}$ for all integers $n \geq k$.

Proposed by: Joe Benton and Dominic Yeo, United Kingdom