BALTIC-WAY 2011 Problem 1

The real numbers $x_1,\ldots ,x_{2011}$ satisfy $$x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}'$$ where $x_1',x_2',\ldots,x_{2011}'$ is a permutation of $x_1,x_2,\ldots,x_{2011}$ . Prove that $x_1=x_2=\ldots =x_{2011}$ .