MathNet 2009-part-04 — 175 problems (EN–VI) Question 141

The integers $a, b, c$ satisfy the following condition: $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$. Prove that the number $a^2 + b^2 + c^2$ is a perfect square of some integer.