$n$ is a positive integer. Is the following statement true?
For any $3n$ integers, saying $\{b_1,..,b_{3n}\}$. There exists $n$ of them, saying $\{a_1,..,a_n\}$,so that $\forall$ $1\leq i,j,k\leq n$ we have $a_i+a_j\neq a_k$. Here $i,j,k$ is not necessary to be different.
I tried to mod them by $3$ and use Principle of tolerance, but it does not help when most of them divided by $3$. I also get some easy inequality by considering some possible trivial condition, for example if $b_l$ are ranked from small to big, if $2b_m\geq b_{m+n}$ for some $m$. I can take $\{b_m,..,b_{m+n-1}\}$ as desired.
Any help will be appreciated.