MathNet 2017-part-06 — 176 problems (EN–VI) Question 25

Let $a$, $b$, $c$ be nonzero integers, with $1$ as their only positive common divisor, such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$. Find the number of such triples $(a, b, c)$ with $50 \geq |a| \geq |b| \geq |c| \geq 1$.