Question 160
On the Cartesian coordinate system $Oxy$, consider a sequence of points $A_{n}(x_{n}, y_{n})$ in which $(x_{n})_{n=1}^{\infty}, (y_{n})_{n=1}^{\infty}$ are two sequences of positive numbers satisfying the following conditions:
$$
x_{n+1} = \sqrt{\frac{x_{n}^{2} + x_{n+2}^{2}}{2}}, \quad y_{n+1} = \left(\frac{\sqrt{y_{n}} + \sqrt{y_{n+2}}}{2}\right)^{2} \quad \forall n \geq 1
$$
Suppose that $O, A_{1}, A_{2016}$ belong to a line $d$ and $A_{1}, A_{2016}$ are distinct. Prove that all the points $A_{2}, A_{3}, \ldots, A_{2015}$ lie on one side of $d$.
$$
x_{n+1} = \sqrt{\frac{x_{n}^{2} + x_{n+2}^{2}}{2}}, \quad y_{n+1} = \left(\frac{\sqrt{y_{n}} + \sqrt{y_{n+2}}}{2}\right)^{2} \quad \forall n \geq 1
$$
Suppose that $O, A_{1}, A_{2016}$ belong to a line $d$ and $A_{1}, A_{2016}$ are distinct. Prove that all the points $A_{2}, A_{3}, \ldots, A_{2015}$ lie on one side of $d$.