CMO 2014 Problem 3

Let $p$ be a fixed odd prime. A $p$-tuple $\left(a_{1}, a_{2}, a_{3}, \ldots, a_{p}\right)$ of integers is said to be good if

(i) $0 \leq a_{i} \leq p-1$ for all $i$, and

(ii) $a_{1}+a_{2}+a_{3}+\cdots+a_{p}$ is not divisible by $p$, and

(iii) $a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}+\cdots+a_{p} a_{1}$ is divisible by $p$.

Determine the number of good $p$-tuples.