Non-zero real numbers $a$, $b$ and $c$ are such that any two of the three equations $ax^{11}+bx^4+c=0$, $bx^{11}+cx^4+a=0$, $cx^{11}+ax^4+b=0$ have a common root. Prove that all three equations have a common root. (Author: I. Bogdanov)