MathDB

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

Problem Statement