APICS Mathematics Contest 1990


  1. Determine with proof whether is convergent or divergent.

  2. Find a formula for and prove it is valid.

  3. Define a sequence by

    Find with justification .

  4. (a) Find with proof a pair of invertible 2x2 real matricies A and B such that all non-trivial linear combinations of A and B are also invertible.
    (b) Can you solve the problem of (a) for 3x3 maticies A? Why(not)?

  5. Show that the triangle formed by a tangent to a hyperbola and its two asymptotes has constant area.

  6. Determine all real numbers m such that the equation

    has 4 real roots in arithmetic progression.

  7. Let be n positive numbers and any permutation of them. Show that

  8. Suppose that a real valued function satisfies

    for some fixed p > 1 and all real x,y. Show that f is a constant function.