CANADA_MO/md/en-exam1996.md

Canadian Mathematical Olympiad 1996

Problem 1

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

$$\frac{1+\alpha }{1-\alpha }+\frac{1+\beta }{1-\beta }+\frac{1+\gamma }{1-\gamma }\backslash frac\{1+\backslash alpha\}\{1-\backslash alpha\}+\backslash frac\{1+\backslash beta\}\{1-\backslash beta\}+\backslash frac\{1+\backslash gamma\}\{1-\backslash gamma\}$$

PROBLEM 2

Find all real solutions to the following system of equations. Carefully justify your answer.

$$\{\begin{array}{l}\frac{4x^2}{1+4x^2}=y\\ \frac{4y^2}{1+4y^2}=z\\ \frac{4z^2}{1+4z^2}=x\end{array}\backslash left\backslash \{\backslash begin\{array\}\{l\}\; \backslash frac\{4\; x^\{2\}\}\{1+4\; x^\{2\}\}=y\; \backslash \backslash \; \backslash frac\{4\; y^\{2\}\}\{1+4\; y^\{2\}\}=z\; \backslash \backslash \; \backslash frac\{4\; z^\{2\}\}\{1+4\; z^\{2\}\}=x\; \backslash end\{array\}\backslash right.$$

PROBLEM 3 We denote an arbitrary permutation of the integers $1, \ldots, n$ by $a_{1}, \ldots, a_{n}$. Let $f(n)$ be the number of these permutations such that (i) $a_{1}=1$; (ii) $\left|a_{i}-a_{i+1}\right| \leq 2, \quad i=1, \ldots, n-1$.

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

PROblem 4

Let $\triangle A B C$ be an isosceles triangle with $A B=A C$. Suppose that the angle bisector of $\angle B$ meets $A C$ at $D$ and that $B C=B D+A D$. Determine $\angle A$.

Problem 5 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 $f(n)=n-\sum_{k=1}^{m}\left[r_{k} n\right]$ 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 $x$