Question 156
Are there sequences of nonnegative integers $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ satisfying the following property:
$$
a_0 = b_0 = 0 \text{ and for any } k \ge 1
$$
(i) $a_k$ is the smallest positive integer that cannot be written as $a_i + b_j$ for some $0 \le i \le k-1$ and $j \ge 0$, and
(ii) $b_k$ is the smallest positive integer that cannot be written as $a_i + b_j$ for some $i \ge 0$ and $0 \le j \le k-1$, and
(iii) The number $2022$ can be written uniquely as $a_i + b_j$ for some $i \ge 0$ and $j \ge 0$.
$$
a_0 = b_0 = 0 \text{ and for any } k \ge 1
$$
(i) $a_k$ is the smallest positive integer that cannot be written as $a_i + b_j$ for some $0 \le i \le k-1$ and $j \ge 0$, and
(ii) $b_k$ is the smallest positive integer that cannot be written as $a_i + b_j$ for some $i \ge 0$ and $0 \le j \le k-1$, and
(iii) The number $2022$ can be written uniquely as $a_i + b_j$ for some $i \ge 0$ and $j \ge 0$.