2021 IMO 第 5 题

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$ th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$ .

Prove that there exists a value of $k$ such that, on the $k$ th move, Jumpy swaps some walnuts $a$ and $b$ such that $a < k < b$ .

两只松鼠 Bushy 和 Jumpy 收集了 2021 年过冬的核桃。 Jumpy 将核桃从 1 编号到 2021 年，并在他们最喜欢的树周围的地面上挖了 2021 个圆形小洞。第二天早上，跳跃注意到布希在每个洞里都放了一个核桃，但没有注意编号。跳跳不高兴，决定通过执行一系列 2021 年的动作来重新排列核桃。在第 $k$ 步中，Jumpy 交换了与核桃 $k$ 相邻的两个核桃的位置。

证明存在 $k$ 的值，使得在第 $k$ 步时，Jumpy 交换一些核桃 $a$ 和 $b$ ，使得 $a < k < b$ 。