Question 74
Let $p$ be a fixed odd prime. A $p$-tuple $(a_1, a_2, a_3, \dots, a_p)$ of integers is said to be good if
(i) $0 \le a_i \le p-1$ for all $i$, and
(ii) $a_1 + a_2 + a_3 + \dots + a_p$ is not divisible by $p$, and
(iii) $a_1a_2 + a_2a_3 + a_3a_4 + \dots + a_p a_1$ is divisible by $p$.
Determine the number of good $p$-tuples.
(i) $0 \le a_i \le p-1$ for all $i$, and
(ii) $a_1 + a_2 + a_3 + \dots + a_p$ is not divisible by $p$, and
(iii) $a_1a_2 + a_2a_3 + a_3a_4 + \dots + a_p a_1$ is divisible by $p$.
Determine the number of good $p$-tuples.