Question 123
An array of $N$ ($N \ge 3$) integer numbers is written on a blackboard. Per move one changes the array using the following rule: let $s$ be the arithmetic mean of all numbers of the current array, then each number which is less than or equal to $s-1$ is increased by 1 and each number which is greater than or equal to $s+1$ is decreased by 1; all other numbers are not changed.
Prove that after a finite number of moves the numbers on the blackboard cannot be changed.
(E. Zhibrik)
Prove that after a finite number of moves the numbers on the blackboard cannot be changed.
(E. Zhibrik)