How to find an array of probabilities that give equal products
Let the array be $a_1,\ldots,a_n$, which we assume are positive. The condition $a_i x_i = a_j x_j$ implies that
$$ x_i = \frac{a_n}{a_i} x_n. $$
Since the $x_i$ must sum to $1$,
$$
1 = \sum_{i=1}^n \frac{a_n}{a_i} x_n \Longrightarrow x_n = \frac{\frac{1}{a_n}}{\sum_{i=1}^n \frac{1}{a_i}}.
$$
The formula for $x_i$ shows that
$$
x_j = \frac{\frac{1}{a_j}}{\sum_{i=i}^n \frac{1}{a_i}}.
$$