Question 165
Given an integer $n \ge 2$. Find the minimum real number $\lambda$ such that for any real numbers $a_1, a_2, \dots, a_n$, and $b$, the following inequality holds:
$$
\lambda \sum_{i=1}^{n} \sqrt{|a_i - b|} + \sqrt{n \left| \sum_{i=1}^{n} a_i \right|} \ge \sum_{i=1}^{n} \sqrt{|a_i|}.
$$
$$
\lambda \sum_{i=1}^{n} \sqrt{|a_i - b|} + \sqrt{n \left| \sum_{i=1}^{n} a_i \right|} \ge \sum_{i=1}^{n} \sqrt{|a_i|}.
$$