IOM 2021 Problem 3

Let $a_1,a_2,\ldots,a_n$ ( $n\geq 2$ ) be nonnegative real numbers whose sum is $\frac{n}{2}$ . For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$ . Prove that $b_i\geq 1$ holds for at least one index $i$ .