CGMO 2006 Problem 8

Let $p$ be a prime number that is greater than $3$ . Show that there exist some integers $a_{1}, a_{2}, \cdots a_{k}$ that satisfy: $$-\frac{p}{2}< a_{1}< a_{2}< \cdots <a_{k}< \frac{p}{2}$$ making the product: $$\frac{p-a_{1}}{|a_{1}|}\cdot \frac{p-a_{2}}{|a_{2}|}\cdots \frac{p-a_{k}}{|a_{k}|}$$ equals to $3^{m}$ where $m$ is a positive integer.