HK-PC 2017

Consider the sequences with 2016 terms formed by the digits 1, 2, 3, and 4. Find the number of those sequences containing an even number of the digit 1.

Find the first digit after the decimal point of the number $\frac1{1009}+\frac1{1010}+\cdots + \frac1{2016}$

Given infinite sequences $a_1,a_2,a_3,\cdots$ and $b_1,b_2,b_3,\cdots$ of real numbers satisfying $a_{n+1}+b_{n+1}=\frac{a_n+b_n}{2}$ and $a_{n+1}b_{n+1}=\sqrt{a_nb_n}$ for all $n\geq1$ . Suppose $b_{2016}=1$ and $a_1>0$ . Find all possible values of $a_1$