Question 121
Let $p$ be a prime number greater than $3$. Prove that there exist integers $a_1, a_2, \dots, a_t$ that satisfy the following conditions:
a.
$$
-\frac{p}{2} \lt a_1 \lt a_2 \lt \cdots \lt a_t \le \frac{p}{2},
$$
b.
$$
\frac{p-a_1}{|a_1|} \cdot \frac{p-a_2}{|a_2|} \cdot \cdots \cdot \frac{p-a_t}{|a_t|} = 3^m,
$$
where $m$ is a positive integer.
a.
$$
-\frac{p}{2} \lt a_1 \lt a_2 \lt \cdots \lt a_t \le \frac{p}{2},
$$
b.
$$
\frac{p-a_1}{|a_1|} \cdot \frac{p-a_2}{|a_2|} \cdot \cdots \cdot \frac{p-a_t}{|a_t|} = 3^m,
$$
where $m$ is a positive integer.