KöMaL 2005 A374

The real numbers $x_1 \ge x_2 \ge$...$\ge x_n$>0 and $y_1 \ge y_2 \ge$...$\ge y_n$>0 satisfy $\prod_{i=1}^kx_i\ge\prod_{i=1}^ky_i.$ for all 0<$k \le n$. Prove that $\sum_{i=1}^nx_i\ge\sum_{i=1}^ny_i.$