CMO 1996

If $\alpha$ , $\beta$ , and $\gamma$ are the roots of $x^3 - x - 1 = 0$ , compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$ .

Find all real solutions to the following system of equations. Carefully justify your answer. $$\left\{ \begin{array}{c} \frac{4x^2}{1+4x^2} = y \frac{4y^2}{1+4y^2} = z \frac{4z^2}{1+4z^2} = x \end{array} \right.$$

We denote an arbitrary permutation of the integers $1$ , $2$ , $\ldots$ , $n$ by $a_1$ , $a_2$ , $\ldots$ , $a_n$ . Let $f(n)$ denote the number of these permutations such that:

(1) $a_1 = 1$ ;

(2): $|a_i - a_{i+1}| \leq 2$ , $i = 1, \ldots, n - 1$ .

Determine whether $f(1996)$ is divisible by 3.

Let $r_1$ , $r_2$ , $\ldots$ , $r_m$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^m r_k = 1$ . Define the function $f$ by $f(n)= n-\sum_{k=1}^m \: [r_k n]$ for each positive integer $n$ . Determine the minimum and maximum values of $f(n)$ . Here ${\ [ x ]}$ denotes the greatest integer less than or equal to $x$ .