1969 Canadian MO Problems
Show that if and are not all zero, then for every positive integer
Determine which of the two numbers , is greater for any .
Let be an equilateral triangle, and be an arbitrary point within the triangle. Perpendiculars are drawn to the three sides of the triangle. Show that, no matter where is chosen, .
Let be a triangle with sides of length , and . Let the bisector of the cut at . Prove that the length of is
Find the sum of , where .
Show that there are no integers for which .
Let be a function with the following properties:
1) is defined for every positive integer ;
2) is an integer;
4) for all and ;
5) whenever .
Prove that .
Show that for any quadrilateral inscribed in a circle of radius the length of the shortest side is less than or equal to .
Let be the right-angled isosceles triangle whose equal sides have length 1. is a point on the hypotenuse, and the feet of the perpendiculars from to the other sides are and . Consider the areas of the triangles and , and the area of the rectangle . Prove that regardless of how is chosen, the largest of these three areas is at least .