BALTIC-WAY 2010 Problem 1

Find all quadruples of real numbers $(a, b, c, d)$ satisfying the system of equations

$$\left\{\begin{array}{l} (b+c+d)^{2010}=3 a \\ (a+c+d)^{2010}=3 b \\ (a+b+d)^{2010}=3 c \\ (a+b+c)^{2010}=3 d \end{array}\right.$$