Question 160
Prove that
$$
\log_{2016} \sqrt[2016]{1+\frac{a_{2}}{a_{1}}} + \log_{2016} \sqrt[2016]{1+\frac{a_{3}}{a_{2}}} + \cdots + \log_{2016} \sqrt[2016]{1+\frac{a_{1}}{a_{2016}}} \geq \frac{1}{5+\log_{2} 63}
$$
for all strictly positive real numbers $a_{1}, a_{2}, \ldots, a_{2016}$.
$$
\log_{2016} \sqrt[2016]{1+\frac{a_{2}}{a_{1}}} + \log_{2016} \sqrt[2016]{1+\frac{a_{3}}{a_{2}}} + \cdots + \log_{2016} \sqrt[2016]{1+\frac{a_{1}}{a_{2016}}} \geq \frac{1}{5+\log_{2} 63}
$$
for all strictly positive real numbers $a_{1}, a_{2}, \ldots, a_{2016}$.