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 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

3. 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-1BA = .

(b)
Prove or disprove: the determinant of A + I is real.

4. Let f (x) be continuously differentiable on (0,) and suppose limx - > f'(x) = 0. Prove that limx - > 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 = 1an diverges. Determine whether Sn = 1an/Sn2 converges, where Sn = a1 + a2 + ... + an.

7. Let S be a finite set of non-negative 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 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 S and 1kn. 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.
(a)
Find d2({1, x, x + 1, y}).

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

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

Peter Linnell
2001-08-12