Nemotron-IMO-Bench 2026 N5

Let $p>3$ be a prime and for each $k\in \{1,2,...,p-1\}$ let $I(k)$ be the unique integer in $\{1,...,p-1\}$ such that $kI(k)\equiv 1\pmod p$. Prove that there are more than $p/4-1$ integers $k\in \{1,...,p-2\}$ such that $I(k+1)<I(k)$.