1969 Canadian MO Problems
Contents
Problem 1
Show that if and are not all zero, then for every positive integer
Problem 2
Determine which of the two numbers , is greater for any .
Problem 3
Let be the length of the hypotenuse of a right triangle whose two other sides have lengths and . Prove that . When does the equality hold?
Problem 4
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, .
Problem 5
Let be a triangle with sides of length , and . Let the bisector of the cut at . Prove that the length of is
Problem 6
Find the sum of , where .
Problem 7
Show that there are no integers for which .
Problem 8
Let be a function with the following properties:
1) is defined for every positive integer ;
2) is an integer;
3) ;
4) for all and ;
5) whenever .
Prove that .
[[[1969 Canadian MO Problems/Problem 8 | Solution]]
Problem 9
Show that for any quadrilateral inscribed in a circle of radius the length of the shortest side is less than or equal to .
Problem 10
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 .