Question 59
Find the least positive real number $\lambda$ such that, for any real numbers $a_1, a_2, a_3 \in [0, \frac{1}{2}]$ and $b_1, b_2, b_3 \in (0, \infty)$ with $\sum_{i=1}^3 a_i = \sum_{i=1}^3 b_i = 1$, the following inequality holds
$$
b_1b_2b_3 \le \lambda(a_1b_1 + a_2b_2 + a_3b_3).
$$
$$
b_1b_2b_3 \le \lambda(a_1b_1 + a_2b_2 + a_3b_3).
$$