CN-SMO 2005 Problem 6

Let $P(A)$ be the arithmetic-means of all elements of set $A = \{ a_1, a_2, \ldots, a_n \}$ , namely $P(A) = \frac{1}{n} \sum^{n}_{i=1}a_i$ . We denote $B$ "balanced subset" of $A$ , if $B$ is a non-empty subset of $A$ and $P(B) = P(A)$ . Let set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$ .

Find the number of all "balanced subset" of $M$ .