1969 Canadian MO Problems

Problem 1

Show that if $\displaystyle a_1/b_1=a_2/b_2=a_3/b_3$ and $\displaystyle p_1,p_2,p_3$ are not all zero, then $\displaystyle\left(\frac{a_1}{b_1} \right)^n=\frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}$ for every positive integer $\displaystyle n.$

Problem 2

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

Problem 3

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

Problem 4

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

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

1) $\displaystyle f(n)$ is defined for every positive integer $\displaystyle n$ ;

2) $\displaystyle f(n)$ is an integer;

3) $\displaystyle f(2)=2$ ;

4) $\displaystyle f(mn)=f(m)f(n)$ for all $\displaystyle m$ and $\displaystyle n$ ;

5) $\displaystyle f(m)>f(n)$ whenever $m>n$ .

Prove that $\displaystyle f(n)=n$ .

Problem 9

Show that for any quadrilateral inscribed in a circle of radius $\displaystyle 1,$ the length of the shortest side is less than or equal to $\displaystyle \sqrt{2}$ .

Problem 10

Let $\displaystyle ABC$ be the right-angled isosceles triangle whose equal sides have length 1. $\displaystyle P$ is a point on the hypotenuse, and the feet of the perpendiculars from $\displaystyle P$ to the other sides are $\displaystyle Q$ and $\displaystyle R$ . Consider the areas of the triangles $\displaystyle APQ$ and $\displaystyle PBR$ , and the area of the rectangle $\displaystyle QCRP$ . Prove that regardless of how $\displaystyle P$ is chosen, the largest of these three areas is at least $\displaystyle 2/9$ .

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