Question 69
Let $n$ be a positive integer and let $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ be positive real numbers. Prove that
$$
a_1 \sqrt{\frac{a_1}{b_1}} + a_2 \sqrt{\frac{a_2}{b_2}} + \dots + a_n \sqrt{\frac{a_n}{b_n}} \ge (a_1 + a_2 + \dots + a_n) \sqrt{\frac{a_1 + a_2 + \dots + a_n}{b_1 + b_2 + \dots + b_n}}
$$
$$
a_1 \sqrt{\frac{a_1}{b_1}} + a_2 \sqrt{\frac{a_2}{b_2}} + \dots + a_n \sqrt{\frac{a_n}{b_n}} \ge (a_1 + a_2 + \dots + a_n) \sqrt{\frac{a_1 + a_2 + \dots + a_n}{b_1 + b_2 + \dots + b_n}}
$$