Question 72
Let $p, n$ be positive integers such that $p$ is prime and $p \lt n$. If $p$ divides $n+1$ and $\left(\left[\frac{n}{p}\right], (p-1)!\right) = 1$, then prove that $p \cdot \left[\frac{n}{p}\right]^2$ divides $\binom{n}{p} - \left[\frac{n}{p}\right]$.
(Here $[x]$ represents the integer part of the real number $x$.)
(Here $[x]$ represents the integer part of the real number $x$.)