Question 64
Let $a_0, a_1, \dots, a_{100}$ and $b_1, b_2, \dots, b_{100}$ be two real sequences such that for all $n = 0, 1, \dots, 99$
$$
a_{n+1} = \frac{a_n}{2}, \quad b_{n+1} = \frac{1}{2} - a_n \quad \text{or} \quad a_{n+1} = 2a_n^2, \quad b_{n+1} = a_n
$$
holds. If $a_{100} \le a_0$, find the maximal value of $b_1 + b_2 + \dots + b_{100}$.
$$
a_{n+1} = \frac{a_n}{2}, \quad b_{n+1} = \frac{1}{2} - a_n \quad \text{or} \quad a_{n+1} = 2a_n^2, \quad b_{n+1} = a_n
$$
holds. If $a_{100} \le a_0$, find the maximal value of $b_1 + b_2 + \dots + b_{100}$.