tmua.me Tau Paper 2 · Question 16

Let

R(x)=\frac{x^3-3x}{3x^2-1}.

A real number $a$ is called admissible if the sequence $x_1=a$ , $x_{n+1}=R(x_n)$ is defined and positive for every positive integer $n$ , and has least positive period $12$ .

In addition, among $x_1,\ldots,x_{12}$ exactly six terms exceed $\sqrt3$ , and exactly two of the twelve cyclic adjacent pairs

(x_1,x_2),(x_2,x_3),\ldots,(x_{11},x_{12}),(x_{12},x_1)

have both terms exceeding $\sqrt3$ .

How many admissible values of $a$ are there?

You may use the identity

R\left(\frac{\cos\theta}{\sin\theta}\right) =\frac{\cos3\theta}{\sin3\theta}

whenever both sides are defined.

276

288

300

600

144

examday

tmua.me Tau Paper 2 · Q16