KZ-MO 1999 Problem 5

For real numbers $x_1, x_2, \dots, x_n$ and $y_1, y_2, \dots, y_n$ , the inequalities hold $x_1 \geq x_2 \geq \ldots \geq x_n> 0$ and $$y_1 \geq x_1, ~ y_1y_2 \geq x_1x_2, ~ \dots, ~ y_1y_2 \dots y_n \geq x_1x_2 \dots x_n.$$ Prove that $ny_1 + (n-1) y_2 + \dots + y_n \geq x_1 + 2x_2 + \dots + nx_n$ .