1969 Canadian MO Problems

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 $\left(\frac{a_1}{b_1} \right)^n=\frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}$ 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 $c\ge 1$.


Problem 3

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


Problem 4

Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, $\frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}$.


Problem 5

Let $ABC$ be a triangle with sides of length $a$, $b$ and $c$. Let the bisector of the $\angle C$ cut $AB$ at $D$. Prove that the length of $CD$ is $\frac{2ab\cos \frac{C}{2}}{a+b}.$


Problem 6

Find the sum of $1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!$, where $n!=n(n-1)(n-2)\cdots2\cdot1$.


Problem 7

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


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) $f(mn)=f(m)f(n)$ 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 $ABC$ 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 $APQ$ and $PBR$, and the area of the rectangle $QCRP$. Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2/9$.