Question 153
Positive numbers $x_1, x_2, \dots, x_{2009}$ satisfy the relations $x_1^2 - x_1x_2 + x_2^2 = x_2^2 - x_2x_3 + x_3^2 = x_3^2 - x_3x_4 + x_4^2 = \dots = x_{2008}^2 - x_{2008}x_{2009} + x_{2009}^2 = x_{2009}^2 - x_{2009}x_1 + x_1^2$. Prove that $x_1 = x_2 = \dots = x_{2009}$.