There was this question posed in IMO 1983:
"Can you choose 1983 pairwise distinct positive integers less than $10^5$, such that no three are in Arithmetic Progression?"
A close look at the sequence reveals the following things as mentioned in Problem Solving Strategies by Arthur Engel.
When the numbers which belong to the sequence are written in the ternary system, all the digits in any of the number is either 0 or 1
A further look suggests that the $nth$ term of the sequence has the same representation in the ternary system as $n$ has in the binary system
These help me to solve the original problem However, I am unsure as to how to prove the above two points
Any help please!!