19. Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1 .
19. Answer: 2010
Consider any integer $k$ where $1 \leq k \leq 2010$. By division algorithm, there exists unique pair of integers $(q, r)$ such that $2010=k q+r$ with $0 \leq r \leq k-1$. We rewrite this as $2010=(k-r) q+r(q+1)$. That is, $k-r$ copies of $q$ and $r$ copies of $q+1$ add up to 2010. Thus there is one desired expression for each value of $k$, which is clearly unique. Hence there are 2010 such expressions in all.