Question 75
Determine whether or not there exist two different sets $A$, $B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 \lt x \lt 1$ satisfies the following inequality:
$$
\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| \lt (1-x)^{2011}.
$$
$$
\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| \lt (1-x)^{2011}.
$$