Question 191
Consider a positive integer $n$ that can be expressed as the sum of positive integers $a_1 \gt a_2 \gt \cdots \gt a_m$ such that
$$
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_m} = 1,
$$
where $m \ge 2$. Prove that $n + 7 \le a_1 a_2$ and determine the cases when equality holds.
$$
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_m} = 1,
$$
where $m \ge 2$. Prove that $n + 7 \le a_1 a_2$ and determine the cases when equality holds.