AUSTRIAN-POLISH 2003 Problem 8

Given reals $x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0$ , show that $$x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$ for any positive integer $n$ .