# 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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