2021 IMO Shortlist N2

Let $n \geq 100$ be an integer. The numbers $n, n+1, \ldots, 2 n$ are written on $n+1$ cards, one number per card. The cards are shuffled and divided into two piles. Prove that one of the piles contains two cards such that the sum of their numbers is a perfect square.

设 $n \geq 100$ 为整数。数字 $n、n+1、\ldots、2 n$ 写在 $n+1$ 卡片上，每张卡片一个数字。将牌洗好并分成两堆。证明其中一堆包含两张牌，且它们的数字之和为完全平方数。