Problem

Source: Turkey Team Selection Test 2018 P6

Tags: algebra, Sequence



$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either $$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$or $$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$Given $a_{100}\leq a_0$, what is the maximal value of $b_1+b_2+\cdots+b_{100}$?