Question 122
Determine all positive integers $n$ satisfying the following condition: There exist pairwise distinct integers $a_1, \dots, a_n, b_1, \dots, b_n$ such that
$$
\prod_{i=1}^{n} (a_k^2 + a_i a_k + b_i) = \prod_{i=1}^{n} (b_k^2 + a_i b_k + b_i) = 0 \quad \text{for all } k = 1, \dots, n.
$$
$$
\prod_{i=1}^{n} (a_k^2 + a_i a_k + b_i) = \prod_{i=1}^{n} (b_k^2 + a_i b_k + b_i) = 0 \quad \text{for all } k = 1, \dots, n.
$$