59th Ukrainian National Mathematical Olympiad

Clearly, the expression cannot equal $0$, since it contains $1009$ odd numbers. We want to show that the expression can be any odd number between $1$ and $1+2+3+\cdots +2018$.

Take some configuration. Find the first from the left consecutive numbers with signs "-" and "+". The switching of these signs decreases the value of the expression by $2$. We will start with the following expression: "+1+2+3+\dots+2018" and will change it to expression "-1+2+3+\dots+2018" that is less by $2$ as described above. By following the described algorithm, we will obtain all the numbers decreased by $2$.

When we get the expression

+1 - 2 - 3 - ... - 2018, for which the algorithm doesn't work, the last step will be to switch to the smallest expression -1 - 2 - 3 - ... - 2018.