CN-TST 2022 Problem 15

Given a positive integer $n \ge 2$ . Find all $n$ -tuples of positive integers $(a_1,a_2,\ldots,a_n)$ , such that $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$ , $a_1$ is odd, and (1) $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$ is a positive integer; (2) One can pick $n$ -tuples of integers $(k_{i,1},k_{i,2},\ldots,k_{i,n})$ for $i=1,2,\ldots,M$ such that for any $1 \le i_1 <i_2 \le M$ , there exists $j \in \{1,2,\ldots,n\}$ such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$ .