, 2014

Solution:

Answer: $1007$

Note that $f(2014)=f(1007)+1$, so there must be exactly one index $1008 \leq i \leq 2014$ such that $f(i)=f(i-1)+1$, and for all $1008 \leq j \leq 2014$, $j \neq i$ we must have $f(j)=f(j-1)$. We first claim that each value of $i$ corresponds to exactly one 2014-tuple $(f(1), \ldots, f(2014))$. To prove this, note that $f(1024)=11$, so each $i$ uniquely determines the values of $f(1007), \ldots, f(2014)$. Then all of $f(1), \ldots, f(1006)$ can be uniquely determined from these values because for any $1 \leq k \leq 1006$, there exists a unique $n$ such that $1007 \leq k \cdot 2^{n} \leq 2014$. It's also clear that these values satisfy the condition that $f$ is nondecreasing, so we have a correspondence from each $1008 \leq i \leq 2014$ to a unique 2014-tuple.

Also, given any valid 2014-tuple $(f(1), \ldots, f(2014))$, we know that $f(1), \ldots, f(1006)$ can be uniquely determined by $f(1007), \ldots, f(2014)$, which yields some $1008 \leq i \leq 2014$ where $f(i)=f(i-1)+1$, so we actually have a bijection between possible values of $i$ and 2014-tuples. Therefore, the total number of possible 2014-tuples is $1007$.