APICS Mathematics Contest 1995


  1. Given the functions and , with for every , and a a real number such that , prove taht there exists exactly one function such that

    for every .

  2. A solid fence encloses a square field with sides of length L. A cow is in a meadow surrounding the field for a large distance on all sides, and is tied to a rope of length R attached to a corner of the fence. What area of the meadow is available for the cow to use?

  3. Find all solutions to

    where x and y are integers different from zero.

  4. For what positive integers n is the n Catalan number,

    odd?

  5. N pairs of diametrically opposite points are chosen on a circle of radius 1. Every line segment joining two of the 2N points, whether in the same pair or not, is called a diagonal. Show that the sum of the squares of the lengths of the diagonals depends only on N; and find that value.

  6. A finite pattern of checkers is placed on an infinite checkerboard, at most one checker to a square; this is Generation 0. Generation N is generated from Generation N-1 (for ) by the following process: if a cell has an odd number of immediate horizontal or vertical neighbours in Generation N-1, it contains a checker in Generation N; otherwise it is vacant.

    Show that there exists an X such that Generation X consists of at least 1995 copies of the original pattern, each separated from the rest of the pattern by an empty region at least 1995 cells wide.

  7. A and B play a game. First A chooses a sequence of three tosses of a coin and tells it to B; then B chooses a different sequence of three tosses and tells it to A. Then they throw a fair coin repeatedly until one sequence or the other shows up as three consecutive tosses.

    For instance, A might choose (head, tail, head); then B might choose (tail, head, tail). If the sequence of tosses is (head, tail, tail, head, tail), B would win.

    If both players play rationally (make their best possible choice), what is the probability that A wins?