Putnam

The answer is $\frac{p-1}{2}$. Define the operator $D = x \frac{d}{dx}$, where $\frac{d}{dx}$ indicates formal differentiation of polynomials. For $n$ as in the problem statement, we have $q(x) = (x-1)^n r(x)$ for some polynomial $r(x)$ in $\mathbb{F}_p$ not divisible by $x-1$. For $m=0,\dots,n$, by the product rule we have $$(D^m q)(x) \equiv n^m x^m (x-1)^{n-m} r(x) \pmod{(x-1)^{n-m+1}}.$$ Since $r(1) \neq 0$ and $n \not\equiv 0 \pmod{p}$ (because $n \leq \deg(q) = p-1$), we may identify $n$ as the smallest nonnegative integer for which $(D^n q)(1) \neq 0$.

Now note that $q = D^{(p-1)/2} s$ for $$s(x) = 1 + x + \cdots + x^{p-1} = \frac{x^p-1}{x-1} = (x-1)^{p-1}$$ since $(x-1)^p = x^p-1$ in $\mathbb{F}_p[x]$. By the same logic as above, $(D^n s)(1) = 0$ for $n=0,\dots,p-2$ but not for $n=p-1$. This implies the claimed result.