Question 121
Let $n \gt 1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \lfloor \frac{n}{2} \rfloor$ such that the following equation has integer solutions with $a_m \gt 0$
$$
\frac{a_m}{m+1} + \frac{a_{m+1}}{m+2} + \dots + \frac{a_{n-1}}{n} = \frac{1}{\text{lcm}(1, 2, \dots, n)}.
$$
$$
\frac{a_m}{m+1} + \frac{a_{m+1}}{m+2} + \dots + \frac{a_{n-1}}{n} = \frac{1}{\text{lcm}(1, 2, \dots, n)}.
$$