Putnam 2025 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 $p/4-1$.