CANADA_MO/md/en-exam1998.md

THE 1998 CANADIAN MATHEMATICAL OLYMPIAD

1. Determine the number of real solutions $a$ to the equation

$$\left[\frac{1}{2}a\right]+\left[\frac{1}{3}a\right]+\left[\frac{1}{5}a\right]=a\backslash left[\backslash frac\{1\}\{2\}\; a\backslash right]+\backslash left[\backslash frac\{1\}\{3\}\; a\backslash right]+\backslash left[\backslash frac\{1\}\{5\}\; a\backslash right]=a$$

Here, if $x$ is a real number, then $[x]$ denotes the greatest integer that is less than or equal to $x$. 2. Find all real numbers $x$ such that

$$x={(x-\frac{1}{x})}^{1\mathrm{/}2}+{(1-\frac{1}{x})}^{1\mathrm{/}2}x=\backslash left(x-\backslash frac\{1\}\{x\}\backslash right)^\{1\; /\; 2\}+\backslash left(1-\backslash frac\{1\}\{x\}\backslash right)^\{1\; /\; 2\}$$

3. Let $n$ be a natural number such that $n \geq 2$. Show that

$$\frac{1}{n+1}(1+\frac{1}{3}+\cdots +\frac{1}{2n-1})>\frac{1}{n}(\frac{1}{2}+\frac{1}{4}+\cdots +\frac{1}{2n})\mathrm{.}\backslash frac\{1\}\{n+1\}\backslash left(1+\backslash frac\{1\}\{3\}+\backslash cdots+\backslash frac\{1\}\{2\; n-1\}\backslash right)>\backslash frac\{1\}\{n\}\backslash left(\backslash frac\{1\}\{2\}+\backslash frac\{1\}\{4\}+\backslash cdots+\backslash frac\{1\}\{2\; n\}\backslash right)\; .$$

4. Let $A B C$ be a triangle with $\angle B A C=40^{\circ}$ and $\angle A B C=60^{\circ}$. Let $D$ and $E$ be the points lying on the sides $A C$ and $A B$, respectively, such that $\angle C B D=40^{\circ}$ and $\angle B C E=70^{\circ}$. Let $F$ be the point of intersection of the lines $B D$ and $C E$. Show that the line $A F$ is perpendicular to the line $B C$.
5. Let $m$ be a positive integer. Define the sequence $a_{0}, a_{1}, a_{2}, \ldots$ by $a_{0}=0, a_{1}=m$, and $a_{n+1}=m^{2} a_{n}-a_{n-1}$ for $n=1,2,3, \ldots$. Prove that an ordered pair $(a, b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation

$$\frac{a^2+b^2}{ab+1}=m^2\backslash frac\{a^\{2\}+b^\{2\}\}\{a\; b+1\}=m^\{2\}$$

if and only if $(a, b)$ is of the form $\left(a_{n}, a_{n+1}\right)$ for some $n \geq 0$.