14th CanMO 1982


1. Given a quadrilateral ABCD and a point P, take A' so that PA' is parallel to AB and of equal length. Similarly take PB', PC', PD' equal and parallel to BC, CD, DA respectively. Show that the area of A'B'C'D' is twice that of ABCD.
2. Show that the roots of x³ - x² - x - 1 are all distinct. If the roots are a, b, c show that (a¹⁹⁸² - b¹⁹⁸²)/(a - b) + (b¹⁹⁸² - c¹⁹⁸²)/(b - c) + (c¹⁹⁸² - a¹⁹⁸²)/(c - a) is an integer.
3. What is the smallest number of points in n-dimensional space Rⁿ such that every point of Rⁿ is an irrational distance from at least one of the points.
4. Show that the number of permutations of 1, 2, ... , n with no fixed points is one different from the number with exactly one fixed point.
5. Let the altitudes of a tetrahedron ABCD be AA', BB', CC', DD' (so that A' lies in the plane BCD and similarly for B', C', D'). Take points A", B", C", D" on the rays AA', BB', CC', DD' respectively so that AA'·AA" = BB'·BB" = CC'·CC" = DD'·DD". Show that the centroid of A"B"C"D" is the same as the centroid of ABCD.

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