Question 23
Let $n \ge 2$ be a given positive integer. Prove that there is an $n$-tuple $(a_1, \dots, a_n)$ of pairwise coprime positive integers, each of which is greater than $1402$, such that
$$
\lfloor \frac{a_1}{a_2} \rfloor + \lfloor \frac{a_2}{a_3} \rfloor + \dots + \lfloor \frac{a_n}{a_1} \rfloor = \lfloor \frac{a_2}{a_1} \rfloor + \lfloor \frac{a_3}{a_2} \rfloor + \dots + \lfloor \frac{a_1}{a_n} \rfloor
$$
Note. By $\lfloor x \rfloor$, we mean the greatest integer that doesn't exceed $x$.
$$
\lfloor \frac{a_1}{a_2} \rfloor + \lfloor \frac{a_2}{a_3} \rfloor + \dots + \lfloor \frac{a_n}{a_1} \rfloor = \lfloor \frac{a_2}{a_1} \rfloor + \lfloor \frac{a_3}{a_2} \rfloor + \dots + \lfloor \frac{a_1}{a_n} \rfloor
$$
Note. By $\lfloor x \rfloor$, we mean the greatest integer that doesn't exceed $x$.