I was communicated the following 1994 Miklos Schweitzer problem:
Is there an ordering of the real numbers such that whenever $x<y<z$ (in this ordering), we have $y \neq (x+z)/2$?
I really have no idea how to approach this problem. It is not really on my top list of priorities right now. I'm just curious about the answer.
1) It is obvious that such an ordering should exist? Why? If such an ordering does exist, is it helpful somehow?
2) If we want to prove that such an ordering does not exist, then what is the right path to the proof?
Looks like the answer is yes, there is such an ordering. An article titled "Chaotic ordering of rationals and reals" appeared in December 2011 issue of the American Mathematical Monthly.
The first page of the article can be seen here: http://www.jstor.org/pss/10.4169/amer.math.monthly.118.10.921
Apparently the question is due to Erdos and Graham.