MathNet 2011 Question 93

Nonzero real numbers $a$, $b$, $c$ are chosen so that each two of three equations $a x^{11} + b x^4 + c = 0$, $b x^{11} + c x^4 + a = 0$, $c x^{11} + a x^4 + b = 0$ share a common root. Prove that all three equations share a common root.

(I. Bogdanov)