KMO 2016 Problem 6

For a positive integer $n$ , there are $n$ positive reals $a_1 \ge a_2 \ge a_3 \cdots \ge a_n$ . For all positive reals $b_1, b_2, \cdots b_n$ , prove the following inequality. $$\frac{a_1b_1+a_2b_2 + \cdots +a_nb_n}{a_1+a_2+ \cdots a_n} \le \text{max}\{ \frac{b_1}{1}, \frac{b_1+b_2}{2}, \cdots, \frac{b_1+b_2+ \cdots +b_n}{n} \}$$