Putnam 1994 A6

Let $f_1, \dots, f_{10}$ be bijections of the set of integers such that for each integer $n$, there is some composition $f_{i_1} \circ f_{i_2} \circ \cdots \circ f_{i_m}$ of these functions (allowing repetitions) which maps 0 to $n$. Consider the set of 1024 functions $$\mathcal{F} = \{f_1^{e_1} \circ f_2^{e_2} \circ \cdots \circ f_{10}^{e_{10}}\},$$ $e_i = 0$ or 1 for $1 \leq i \leq 10$. ($f_i^0$ is the identity function and $f_i^1 = f_i$.) Show that if $A$ is any nonempty finite set of integers, then at most 512 of the functions in $\mathcal{F}$ map $A$ to itself.