Question 171
Let $m$ and $n$ be integers such that $m \ge 2$ and $n \ge 3$. Prove that there exist $m$ distinct positive integers $a_1, a_2, a_3, \dots, a_m$, all divisible by $n-1$, such that
$$
\frac{1}{n} = \frac{1}{a_1} - \frac{1}{a_2} + \frac{1}{a_3} - \dots + (-1)^{m-1} \frac{1}{a_m}.
$$
$$
\frac{1}{n} = \frac{1}{a_1} - \frac{1}{a_2} + \frac{1}{a_3} - \dots + (-1)^{m-1} \frac{1}{a_m}.
$$