2025 CMO 第 5 题

Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$ , $|f(a) - f(b)|\leq 2024$ if $p \mid a^2 - b$ . Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$ .

设 $p$ 为素数，$f$ 为从 $\left\{0,1,\ldots,p-1\right\}$ 到自身的双射。假设对于整数 $a,b \in \left\{0,1,\ldots,p-1\right\}$ ， $|f(a) - f(b)|\leq 2024$ if $p \mid a^2 - b$ 。证明存在无限多个 $p$ 使得存在这样的 $f$ ，并且也存在无限多个 $p$ 使得不存在这样的 $f$ 。