9th CanMO 1977


1. Show that there are no positive integers m, n such that 4m(m+1) = n(n+1).
2. X is a point inside a circle center O other than O. Which points P on the circle maximise ∠OPX?
3. Find the smallest positive integer b for which 7 + 7b + 7b² is a fourth power.
4. The product of two polynomials with integer coefficients has all its coefficients even, but at least one not divisible by 4. Show that one of the two polynomials has all its coefficients even and that the other has at least one odd coefficient.
5. A right circular cone has base radius 1. The vertex is K. P is a point on the circumference of the base. The distance KP is 3. A particle travels from P around the cone and back by the shortest route. What is its minimum distance from K?
6. The real sequence x₁, x₂, x₃, ... is defined by x₁ = 1 + k, x_n+1 = 1/x_n + k, where 0 < k < 1. Show that every term exceeds 1.
7. Given m+1 equally spaced horizontal lines and n+1 equally spaced vertical lines forming a rectangular grid with (m+1)(n+1) nodes. Let f(m, n) be the number of paths from one corner to the opposite corner along the grid lines such that the path does not visit any node twice. Show that f(m, n) ≤ 2^mn.

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