1st CanMO 1969


1. a, b, c, d, e, f are reals such that a/b = c/d = e/f; p, q, r are reals, not all zero; and n is a positive integer. Show that (a/b)ⁿ = (p aⁿ + q cⁿ + r eⁿ)/(p bⁿ + q dⁿ + r fⁿ ).
2. If x is a real number not less than 1, which is larger: √(x+1) - √x or √x - √(x-1)?
3. A right-angled triangle has longest side c and other side lengths a and b. Show that a + b ≤ c√2. When do we have equality?
4. The sum of the distances from a point inside an equilateral triangle of perimeter length p to the sides of the triangle is s. Show that s √12 = p.
5. ABC is a triangle with \|BC\| = a, \|CA\| = b. Show that the length of the angle bisector of C is (2ab cos C/2)/(a + b).
6. Find 1.1! + 2.2! + ... + n.n! .
7. Show that there are no integer solutions to a² + b² = 8c + 6.
8. f is a function defined on the positive integers with integer values. Given that (1) f(2) = 2, (2) f(mn) = f(m) f(n) for all m,n, and (3) f(m) > f(n) for all m, n such that m > n, show that f(n) = n for all n.
9. Show that the shortest side of a cyclic quadrilateral with circumradius 1 is at most √2.
10. P is a point on the hypoteneuse of an isosceles, right-angled triangle. Lines are drawn through P parallel to the other two sides, dividing the triangle into two smaller triangles and a rectangle. Show that the area of one of these component figures is at least 4/9 of the area of the original triangle.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.