2020 IMO Shortlist N2

For each prime $p$, there is a kingdom of $p$-Landia consisting of $p$ islands numbered $1,2, \ldots, p$. Two distinct islands numbered $n$ and $m$ are connected by a bridge if and only if $p$ divides $\left(n^{2}-m+1\right)\left(m^{2}-n+1\right)$. The bridges may pass over each other, but cannot cross. Prove that for infinitely many $p$ there are two islands in $p$-Landia not connected by a chain of bridges. (Denmark)

对于每个素数 $p$，都有一个 $p$-Landia 王国，由编号为 $1,2、\ldots、p$ 的 $p$ 岛屿组成。当且仅当 $p$ 整除 $\left(n^{2}-m+1\right)\left(m^{2}-n+1\right)$ 时，编号为 $n$ 和 $m$ 的两个不同岛屿通过桥梁连接。桥梁可以相互交叉，但不能交叉。证明对于无穷多个 $p$，$p$-Landia 中有两个岛屿没有通过桥链连接。 (丹麦)