Question 163
Let $p \gt 3$ be a prime. Find the number of ordered sextuples $(a, b, c, d, e, f)$ of positive integers, whose sum is $3p$, and all the fractions
$$
\frac{a+b}{c+d}, \quad \frac{b+c}{d+e}, \quad \frac{c+d}{e+f}, \quad \frac{d+e}{f+a}, \quad \frac{e+f}{a+b}
$$
are integers.
$$
\frac{a+b}{c+d}, \quad \frac{b+c}{d+e}, \quad \frac{c+d}{e+f}, \quad \frac{d+e}{f+a}, \quad \frac{e+f}{a+b}
$$
are integers.