APICS Mathematics Contest 1983

  1. If n is a positive integer, show that

  2. Given numbers x,y,z such that

    Compute .

  3. An ant starts at a point P on the bottom edge of a right circular cylinder of radius R and height H. If the ant makes n complete circuits around the cylinder and finishes at a point at the top edge directly above its starting point, find, with justification, the length of its shortest possible path.

  4. Let f be an integrable function and let . Show that for .

  5. Given a surface S defined by such that (a) the intersection of S with any plane z = constant is the curve xy = constant and (b) the intersection of S with any plane x = constant is the curve = constant. Find the equation of the surface S.

  6. Determine the locus (path) of the point O of intersection of the altitudes (orthocentre) of a triangle ABC, if the locus of vertex A is a line parallel to BC.

  7. Show that is irrational.

  8. Select a non-negative integer n at random. What is the probability that the first digit of is a ``one'' (in base 10 notation)?