IMO 2021 Problem 1

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ 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.

====================================

Proof:

There must be an integer $k$ such that $2k^2-4k$, $2k^2+1$, and $2k^2+4k$ are all within $[n, 2n]$. Any two of them sum up to a perfect square.

Q.E.D.