1st Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 10, 1979
Fill out the individual registration form
 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.)
 Let S be a set which is closed under the binary operation o,
with the following properties:
 (i)
 there is an element e S such that
aoe = eoa = a,
for each a S,
 (ii)

(aob)o(cod )= (aoc)o(bod ), for
all
a, b, c, d S.
Prove or disprove:
 (a)
 o is associative on S
 (b)
 o is commutative on S
 Let A be an n X n nonsingular matrix with complex elements,
and let be its complex conjugate. Let
B = A + I, where I is the n X n identity matrix.
 (a)
 Prove or disprove:
A^{1}BA = .
 (b)
 Prove or disprove: the determinant of
A + I is real.
 Let f (x) be continuously differentiable on
(0,) and suppose
lim_{x  > }f'(x) = 0. Prove that
lim_{x  > }f (x)/x = 0.
 Show, for all positive integers
n = 1, 2,..., that 14 divides
3^{4n + 2} + 5^{2n + 1}.
 Suppose a_{n} > 0 and
S_{n = 1}^{}a_{n} diverges. Determine
whether
S_{n = 1}^{}a_{n}/S_{n}^{2} converges, where
S_{n} = a_{1} + a_{2} + ... + a_{n}.
 Let S be a finite set of nonnegative integers such that
 x  y S whenever x, y S.
 (a)
 Give an example of such a set which contains ten elements.
 (b)
 If A is a subset of S containing more than twothirds of the
elements of S, prove or disprove that every element of S
is the sum or difference of two elements from A.
 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
p_{1}p_{2}...p_{k}, where p_{i} S
and
1kn. Let d_{n}(S) indicate the maximum number of
linearly independent polynomials in
W_{n}(S). For example,
W_{2}({x^{2}, y}) = {x^{2}, y, x^{2}y, x^{4}, y^{2}} and
d_{2}({x^{2}, y}) = 5.
 (a)
 Find
d_{2}({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  > }(logd_{n}(S))/log n.
(
limsup_{n  > }a_{n} = lim_{n  > }(sup{a_{n}, a_{n + 1},...}),
where sup means supremum or least upper bound.)
Peter Linnell
20010812