3rd Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 7, 1981

Fill out the individual registration form

1. The number 248 - 1 is exactly divisible by what two numbers between 60 and 70?

2. For which real numbers b does the function f (x), defined by the conditions f (0) = b and f' = 2f - x, satisfy f (x) > 0 for all x 0?

3. Let A be non-zero square matrix with the property that A3 = 0, where 0 is the zero matrix, but with A being otherwise arbitrary.
(a)
Express (I - A)-1 as a polynomial in A, where I is the identity matrix.

(b)
Find a 3 X 3 matrix satisfying B2 0, B3 = 0.

4. Define F(x) by F(x) = Sn = 0Fnxn (wherever the series converges), where Fn is the nth Fibonacci number defined by F0 = F1 = 1, Fn = Fn - 1 + Fn - 2, n > 1. Find an explicit closed form for F(x).

5. Two elements A, B in a group G have the property ABA-1B = 1, where 1 denotes the identity element in G.
(a)
Show that AB2 = B-2A.

(b)
Show that ABn = B-nA for any integer n.

(c)
Find u and v so that (BaAb)(BcAd) = BuAv.

6. With k a positive integer, prove that (1 - k-2)k1 - 1/k.

7. Let A = {a0, a1,...} be a sequence of real numbers and define the sequence A' = {a0', a1',...} as follows for n = 0, 1,...: a2n' = an, a2n + 1' = an + 1. If a0 = 1 and A' = A, find
(a)
a1, a2, a3 and a4

(b)
a1981

(c)
A simple general algorithm for evaluating an, for n = 0, 1,....

8. Let
(i)
0 < a < 1,

(ii)
0 < Mk + 1 < Mk, for k = 0, 1,...,

(iii)
limk - > Mk = 0.
If bn = Sk = 0an - kMk, prove that limn - > bn = 0.

Peter Linnell
2001-08-12