HMMT 2010 C8

How many functions $f$ from $\{-1005, \ldots, 1005\}$ to $\{-2010, \ldots, 2010\}$ are there such that the following two conditions are satisfied?

• If $a<b$ then $f(a)<f(b)$.
• There is no $n$ in $\{-1005, \ldots, 1005\}$ such that $|f(n)|=|n|$.