(T. Eisenkölbl) Given is a natural number $n \geq 3$. On a board, the numbers $1,2,3, \ldots, n$ are written. In each move, one chooses two numbers and replaces them with their arithmetic mean. This is done until only one number remains on the board.

Determine the smallest integer that can be achieved at the end through a suitable sequence of moves.
(T. Eisenkölbl) The answer is 2 for every n. We can certainly not achieve a number smaller than 2, as 1 occurs only once and as soon as it is used, it produces an arithmetic mean greater than 1.

On the other hand, we can show by induction that with the allowed moves, from the numbers $a$, $a+1, \ldots, a+k$ with $k \geq 2$, the number $a+1$ can be generated.

For the base case with $k=2$, replace $a$ and $a+2$ with $a+1$, then replace $a+1$ and $a+1$ with an $a+1$.

For the induction step $k \rightarrow k+1$, replace $a+1, \ldots, a+k+1$ with $a+2$ and then replace $a$ and $a+2$ with $a+1$.

With $a=1$ and $k=n-1$, we now obtain the desired result.