Open Contests

The largest number on the blackboard cannot increase on any step, because the absolute value of the difference of two nonnegative numbers cannot be greater than the maximum of these two numbers. Since in the beginning all the numbers are different and positive, after the first step the largest possible number is $2011$ and the smallest possible number is $1$. After the second step the largest possible number is $2010$ and hence the number left on the blackboard in the end cannot be larger than $2010$.

The number $2010$ can be left on the blackboard, for example when in the beginning the numbers are written in the order $2012,1,2,3,\dots ,2011$. Then after the first step there are the numbers $2011,1,1,\dots ,1$, and after the second step the numbers $2010,0,0,\dots ,0$. On each following step the number of zeroes decreases by one and in the end only the number $2010$ remains.