International Mathematical Olympiad 2021 Problem 1

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n + 1, \ldots, 2n$ each on different cards. He then shuffles these $n + 1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.