APICS Mathematics Contest 1979

  1. Let be any polynomial which satisfies the equation

    for a fixed . Find .

  2. Show that if n is an integer greater than 1, then is not prime.

  3. Show that any convex polygon with area 1 can be covered by a parallelogram with area less than or equal to 2.

  4. Let be a set of real numbers. For each non-empty subset T of S, we form average of the elements of T. Find the median of the sequence . (The median of a sequence of numbers is the middle value when the numbers are arranged in non-decreasing order, e.g. the median of 2,4,7,9,9 is 7.)

  5. Let S be a finite set consisting of n elements. Find the total number of unordered pairs of nonempty subsets of S, such that . (Unordered means that ). Simplify the result as much as you can.

  6. Prove the following inequality for all integral :

  7. Let be a sequence of non-negative real numbers such that

    (i) for all ;
    (ii) there exists an such that .

    Show that , i.e. the exists and is finite.