Question 33
Let $n$ be a positive integer. Alice writes $n$ real numbers $a_{1}, a_{2}, \ldots, a_{n}$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors (that is, $a_{n}$ is not a neighbor of $a_{1}$, nor vice versa). A number changes sign if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_{1}$ can change sign, across all possible values of $a_{1}, a_{2}, \ldots, a_{n}$ and all possible sequences of moves Alice may make.