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The Fourteenth Canadian Mathematical Olympiads
1982年第十四届加拿大数学奥林匹克
  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.
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