2016 Canadian Math Olympiad Problem 1

2016 Canada MO 1. The integers 1, 2, 3, …, 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.

Advertisements
This entry was posted in Canada, Grade 12 and tagged , , . Bookmark the permalink.

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s