APICS Mathematics Contest 1981


  1. Let be the general term of a sequence and let . Find, with justification, the term if .

  2. Given positive real numbers a,b,c,d such that a + b + c + d = 1, find the largest possible value
    of .

  3. Let the plane be covered by a net of congruent squares and call the vertices of these squares ``lattice points''. Does there exist an equilateral triangle, all of whose vertices are lattice points? Explain.

  4. Let be the determinant of the matrix

    Assuming the limit exists, find .

  5. Let be non-collinear points in the plane and P and Q are points such that

    Show that there exists a point K such that

  6. Let denote the number of integers such that is odd. Show that is always a power of two.

  7. Four flies sit at the corners of a square card table, side a, facing inward. They start simultaneously walking at the same rate, each directing its motion steadily toward the fly on its right.

    Find

    (i) the equation of the path traced by one of the flies;
    (ii) without calculus, the total distance travelled by each fly.