Question 190
Fix an integer $n \ge 2$. Find all $n$-tuples $(a_1, a_2, \dots, a_n)$ of integers satisfying the following two conditions:
(1) $a_1$ is odd, $1 \lt a_1 \le a_2 \le \dots \le a_n$, and $M = \frac{1}{2^n}(a_1 - 1)a_2 \dots a_n$ is an integer; and
(2) there exist $M$ different $n$-tuples $(c_{i,1}, c_{i,2}, \dots, c_{i,n})$ with $i = 1, 2, \dots, M$, such that for all $1 \le i \lt j \le M$, there exists $k \in \{1, 2, \dots, n\}$ such that
$$
c_{i,k} - c_{j,k} \not\equiv -1, 0, 1 \pmod{a_k}.
$$
(1) $a_1$ is odd, $1 \lt a_1 \le a_2 \le \dots \le a_n$, and $M = \frac{1}{2^n}(a_1 - 1)a_2 \dots a_n$ is an integer; and
(2) there exist $M$ different $n$-tuples $(c_{i,1}, c_{i,2}, \dots, c_{i,n})$ with $i = 1, 2, \dots, M$, such that for all $1 \le i \lt j \le M$, there exists $k \in \{1, 2, \dots, n\}$ such that
$$
c_{i,k} - c_{j,k} \not\equiv -1, 0, 1 \pmod{a_k}.
$$