2025 Putnam B5

Let $p$ be a prime number greater than $3$. For each $k \in \{1,\dots,p-1\}$, let $I(k) \in \{1,2,\dots,p-1\}$ be such that $k \cdot I(k) \equiv 1 \pmod{p}$. Prove that the number of integers $k \in \{1,\dots,p-2\}$ such that $I(k+1) < I(k)$ is greater than $p/4-1$.

令$p$ 为大于$3$ 的质数。对于每个 $k \in \{1,\dots,p-1\}$，令 $I(k) \in \{1,2,\dots,p-1\}$ 使得 $k \cdot I(k) \equiv 1 \pmod{p}$。证明满足 $I(k+1) < I(k)$ 的整数 $k \in \{1,\dots,p-2\}$ 的数量大于 $p/4-1$。