# Canadian Mathematical Olympiad 

## Problem 1

Show that if $a_{1} / b_{1}=a_{2} / b_{2}=a_{3} / b_{3}$ and $p_{1}, p_{2}, p_{3}$ are not all zero, then

$$
\left(\frac{a_{1}}{b_{1}}\right)^{n}=\frac{p_{1} a_{1}^{n}+p_{2} a_{2}^{n}+p_{3} a_{3}^{n}}{p_{1} b_{1}^{n}+p_{2} b_{2}^{n}+p_{3} b_{3}^{n}}
$$

for every positive integer $n$.

## PROBLEM 2

Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}, \sqrt{c}-\sqrt{c-1}$ is greater for any $c \geq 1$.

## Problem 3

Let $c$ be the length of the hypotenuse of a right angle triangle whose other two sides have lengths $a$ and $b$. Prove that $a+b \leq \sqrt{2} c$. When does the equality hold?

Problem 4
Let $A B C$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $P D, P E, P F$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen,

$$
\frac{P D+P E+P F}{A B+B C+C A}=\frac{1}{2 \sqrt{3}}
$$

## PROBLEM 5

Let $A B C$ be a triangle with sides of lengths $a, b$ and $c$. Let the bisector of the angle $C$ cut $A B$ in $D$. Prove that the length of $C D$ is

$$
\frac{2 a b \cos \frac{C}{2}}{a+b}
$$

## PROBLEM 6

Find the sum of $1 \cdot 1!+2 \cdot 2!+3 \cdot 3!+\cdots+(n-1)(n-1)!+n \cdot n!$, where $n!=$ $n(n-1)(n-2) \cdots 2 \cdot 1$.

PROBLEM 7
Show that there are no integers $a, b, c$ for which $a^{2}+b^{2}-8 c=6$.

## Problem 8

Let $f$ be a function with the following properties:

1) $f(n)$ is defined for every positive integer $n$;
2) $f(n)$ is an integer;
3) $f(2)=2$;
4) $f(m n)=f(m) f(n)$ for all $m$ and $n$;
5) $f(m)>f(n)$ whenever $m>n$.

Prove that $f(n)=n$.
PROBLEM 9
Show that for any quadrilateral inscribed in a circle of radius 1 , the length of the shortest side is less than or equal to $\sqrt{2}$.

## PRoblem 10

Let $A B C$ be the right-angled isosceles triangle whose equal sides have length $1 . P$ is a point on the hypotenuse, and the feet of the perpendiculars from $P$ to the other sides are $Q$ and $R$. Consider the areas of the triangles $A P Q$ and $P B R$, and the area of the rectangle $Q C R P$. Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2 / 9$.
![](https://cdn.mathpix.com/cropped/2024_11_22_a66f72136a51e76a6969g-2.jpg?height=461&width=472&top_left_y=870&top_left_x=1130)