CMO 2016 Problem 1

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 .