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:

   (i)

   there is an element e $\in$ S such that aoe = eoa = a, for each a $\in$ S,

   (ii)

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

   Prove or disprove:

   (a)

   o is associative on S

   (b)

   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.

   (a)

   Prove or disprove: A^-1BA = $\bar{B}$.

   (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 lim_{x - > $\infty$}f'(x) = 0. Prove that lim_{x - > $\infty$}f (x)/x = 0.

5. Show, for all positive integers n = 1, 2,..., that 14 divides 3^{4n + 2} + 5^{2n + 1}.

6. Suppose a_n > 0 and S_{n = 1}^$\infty$a_n diverges. Determine whether S_{n = 1}^$\infty$a_n/S_n² converges, where S_n = a₁ + a₂ + ... + a_n.

7. Let S be a finite set of non-negative integers such that | x - y| $\in$ S whenever x, y $\in$ S.

   (a)

   Give an example of such a set which contains ten elements.

   (b)

   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 W_n(S) to be the collection of all expressions p₁p₂...p_k, where p_i $\in$ S and 1$\le$k$\le$n. Let d_n(S) indicate the maximum number of linearly independent polynomials in W_n(S). For example, W₂({x², y}) = {x², y, x²y, x⁴, y²} and d₂({x², y}) = 5.

   (a)

   Find d₂({1, x, x + 1, y}).

   (b)

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

   (c)

   Calculate the least upper bound over all such sets of limsup_{n - > $\infty$}(logd_n(S))/log n. ( limsup_{n - > $\infty$}a_n = lim_{n - > $\infty$}(sup{a_n, a_{n + 1},...}), where sup means supremum or least upper bound.)