MEMO 2009 Problem 7

The numbers $0$ , $1$ , $\dots$ , $n$ ( $n \ge 2$ ) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $g(n)$ for every $n$ .