Interesting set/divisibility/counting question

[Chris11]

Suppose that a set S has the property that all of its members are integers and that the average of any 3 of its members is not an integer. What is the maximum number of elements that S may have?

 

[Martin Rattigan]

The average of three members a,b,c is (a+b+c)/3 and is an integer iff a+b+c=0(3).

If any three members are such that a=b=c(3) then a+b+c=0(3). Similarly if a,b,c are in 0,1,-1(3) in some order a+b+c=0(3).

Therefore the max is 2 from each of 2 residue classes, which is achievable e.g.

1,2,4,5