1st Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 10, 1979

Fill out the individual registration form

  1. Show that the right circular cylinder of volume V which has the least surface area is the one whose diameter is equal to its altitude. (The top and bottom are part of the surface.)

  2. Let S be a set which is closed under the binary operation o, with the following properties:
    there is an element e $ \in$ S such that aoe = eoa = a, for each a $ \in$ S,

    (aob)o(cod )= (aoc)o(bod ), for all a, b, c, d $ \in$ S.

    Prove or disprove:

    o is associative on S

    o is commutative on S

  3. Let A be an n X n nonsingular matrix with complex elements, and let $ \bar{A}$ be its complex conjugate. Let B = A$ \bar{A}$ + I, where I is the n X n identity matrix.
    Prove or disprove: A-1BA = $ \bar{B}$.

    Prove or disprove: the determinant of A$ \bar{A}$ + I is real.

  4. Let f (x) be continuously differentiable on (0,$ \infty$) and suppose limx - > $\scriptstyle \infty$f'(x) = 0. Prove that limx - > $\scriptstyle \infty$f (x)/x = 0.

  5. Show, for all positive integers n = 1, 2,..., that 14 divides 34n + 2 + 52n + 1.

  6. Suppose an > 0 and Sn = 1$\scriptstyle \infty$an diverges. Determine whether Sn = 1$\scriptstyle \infty$an/Sn2 converges, where Sn = a1 + a2 + ... + an.

  7. Let S be a finite set of non-negative integers such that | x - y| $ \in$ S whenever x, y $ \in$ S.
    Give an example of such a set which contains ten elements.

    If A is a subset of S containing more than two-thirds of the elements of S, prove or disprove that every element of S is the sum or difference of two elements from A.

  8. Let S be a finite set of polynomials in two variables, x and y. For n a positive integer, define Wn(S) to be the collection of all expressions p1p2...pk, where pi $ \in$ S and 1$ \le$k$ \le$n. Let dn(S) indicate the maximum number of linearly independent polynomials in Wn(S). For example, W2({x2, y}) = {x2, y, x2y, x4, y2} and d2({x2, y}) = 5.
    Find d2({1, x, x + 1, y}).

    Find a closed formula in n for dn({1, x, y}).

    Calculate the least upper bound over all such sets of limsupn - > $\scriptstyle \infty$(logdn(S))/log n. ( limsupn - > $\scriptstyle \infty$an = limn - > $\scriptstyle \infty$(sup{an, an + 1,...}), where sup means supremum or least upper bound.)

Peter Linnell