Question 54
Let $x_1$ be a positive real number and
$$
x_{n+1} = \sqrt{5x_n + 2\sqrt{x_n^2 + 1}} \quad \text{for } n = 1, 2, 3, \dots
$$
Prove that among $x_1, x_2, \dots, x_{2011}$, there are at least 670 irrational numbers.
$$
x_{n+1} = \sqrt{5x_n + 2\sqrt{x_n^2 + 1}} \quad \text{for } n = 1, 2, 3, \dots
$$
Prove that among $x_1, x_2, \dots, x_{2011}$, there are at least 670 irrational numbers.