MathNet 2008-part-03 — 165 problems (EN–VI) Question 65

Let $x + y + z = a$ and $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{a}$, for $x, y, z, a \in \mathbb{R}$. Prove that at least one of $x, y, z$ is equal to $a$.