MathNet 2000 — Math Olympiad Vault Question 20

If $a, b, c, d, e, f$ are real numbers such that $a+b+c+d+e+f=0$ and $a^{3}+b^{3}+c^{3}+d^{3}+e^{3}+f^{3}=0$, prove that $(a+c)(a+d)(a+e)(a+f)=(b+c)(b+d)(b+e)(b+f)$.