Question 157
The integers $1$, $2$, $3$, $\ldots$, $2016$ are written on a board. You can choose any two numbers on the board and replace them with their average. For example, you can replace $1$ and $2$ with $1.5$, or you can replace $1$ and $3$ with a second copy of $2$. After $2015$ replacements of this kind, the board will have only one number left on it.
a. Prove that there is a sequence of replacements that will make the final number equal to $2$.
b. Prove that there is a sequence of replacements that will make the final number equal to $1000$.
a. Prove that there is a sequence of replacements that will make the final number equal to $2$.
b. Prove that there is a sequence of replacements that will make the final number equal to $1000$.