What is the Sum of the Reciprocals of Variables in Solving Cubic Root Equations?

[wendy<3]

Hi, can you help me with this problem?

a = \sqrt[3]{1 - 4b - 4c}
b = \sqrt[3]{1 - 4c - 4a}
c = \sqrt[3]{1 - 4a - 4b}

Find \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}

Thanks

 

[NeroKid]

a^3 = 1 -4b -4c
b^3 = 1- 4c -4a
c^3 = 1- 4a -4b
=> a^3 +4b +4c = b^3 +4c +4a = c^3 +4a + 4b
<=> a^3 + 4b + 4c = b^3 + 4c + 4a and a^3 + 4b + 4c = c^3 +4a +4b
<=> a+b+c = 0 ( I have ignored the conditions a =b or a =c since they are easy)
it should be easy by replacing a + b + c = 0 into those first equations