GULF 2019 Problem 4

Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$ , and for $n \ge 7$ : $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$ . For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$ .

For all integers $n \ge 2$ , let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$ . The minus $-$ denotes the ''remove it if it is there'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$ . However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$ .

1. Determine $S_7,S_8,S_9$ and $S_{10}$ .
2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$ . Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced .
3. Is the set $S_{2019}$ well-balanced? Justify your answer.