MathNet 2016-part-06 — 184 problems (EN–VI) Question 180

Real nonzero numbers $a$, $b$, $c$, $d$ satisfy the conditions $a^3 + b^3 + c^3 + d^3 = 0$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \neq 0$. Prove that $a + b + c + d \neq 0$.