Question 66
Let $a_{0,1}, a_{0,2}, \dots, a_{0,2016}$ be positive real numbers. For $n \ge 0$ and $1 \le k \lt 2016$ set
$$
a_{n+1,k} = a_{n,k} + \frac{1}{2a_{n,k+1}}
$$
and
$$
a_{n+1,2016} = a_{n,2016} + \frac{1}{2a_{n,1}}.
$$
Let
$$
m_n = \max_{1 \le k \le 2016} a_{n,k} \quad \text{for } n \ge 0.
$$
Show that $m_{2016} \gt 44$.
$$
a_{n+1,k} = a_{n,k} + \frac{1}{2a_{n,k+1}}
$$
and
$$
a_{n+1,2016} = a_{n,2016} + \frac{1}{2a_{n,1}}.
$$
Let
$$
m_n = \max_{1 \le k \le 2016} a_{n,k} \quad \text{for } n \ge 0.
$$
Show that $m_{2016} \gt 44$.