Question 139
Find the maximum positive number $M$ such that for every $n \in \mathbb{N}^*$, there are positive numbers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ satisfying
$$
(a) \sum_{k=1}^{n} b_k = 1,\ 2b_k \ge b_{k-1} + b_{k+1},\ k = 2, 3, \dots, n-1,
$$
$$
(b) a_k^2 \le 1 + \sum_{i=1}^{k} a_i b_i,\ k = 1, 2, \dots, n,
$$
$$
(c) a_n = M.
$$
$$
(a) \sum_{k=1}^{n} b_k = 1,\ 2b_k \ge b_{k-1} + b_{k+1},\ k = 2, 3, \dots, n-1,
$$
$$
(b) a_k^2 \le 1 + \sum_{i=1}^{k} a_i b_i,\ k = 1, 2, \dots, n,
$$
$$
(c) a_n = M.
$$