The Famous problem of Antiquity

Algebra Level 5

Given infinite sequences \(a_1,a_2,a_3,\cdots \) and \(b_1,b_2,b_3,\cdots\) of real numbers satisfying

\(\displaystyle a_{n+1}+b_{n+1}=\frac{a_n+b_n}{2}\)

\(\displaystyle a_{n+1}b_{n+1}=\sqrt{a_nb_n}\)

for all \(n \geq 1\).

Suppose \(b_{2016} = 1\) and \(a_1 > 0\).

Find the number of possible values of \(a_1\).

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