Question 122
$2009$ nonnegative integers are arranged on a circle, each number does not exceed $100$. A positive integer $k$ is fixed. By one move, one can choose two neighboring positions on a circle and add $1$ to both numbers in these positions. It is allowed to make at most $k$ moves for each pair of neighboring positions. Find the least value of $k$ such that, from each initial position, one can make all the numbers equal. (I. Bogdanov)