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Consider a sequence of real numbers defined by: \begin{align*} x_{1} & = c \\ x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \text{for all } n \geq 1. \end{align*} Show that if

c

is a positive integer, then

x_{n}

is an integer for all

n \geq 1

. (South Africa)

3

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