Question 38
Written on a blackboard are the $2023$ numbers
$2023, 2023, \ldots, 2023$.
The numbers on the blackboard are now modified, in a sequence of moves. In each move, two numbers on the blackboard—call them $x$ and $y$—are chosen, deleted, and replaced by the single number $\frac{x+y}{4}$. Such moves are carried out until there is only one number left on the blackboard.
Prove that this number is always greater than $1$.
$2023, 2023, \ldots, 2023$.
The numbers on the blackboard are now modified, in a sequence of moves. In each move, two numbers on the blackboard—call them $x$ and $y$—are chosen, deleted, and replaced by the single number $\frac{x+y}{4}$. Such moves are carried out until there is only one number left on the blackboard.
Prove that this number is always greater than $1$.