2023 APMO 第 1 题

Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1,2, \ldots, n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices.

Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

设 $n \geq 5$ 为整数。考虑边长分别为 $1,2、\ldots、n$ 的 $n$ 个正方形。这些正方形排列在平面上，其边平行于 $x$ 和 $y$ 轴。假设没有两个正方形接触，除非它们的顶点可能接触。

证明可以将这些正方形排列成每个正方形恰好接触其他两个正方形。