RO-TST 1990 Problem 6

Prove that there are infinitely many n's for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$ , and prove that there are infinitely many $n$ 's for which there is no such partition.