3rd Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 7, 1981
Fill out the individual registration form
 The number
2^{48}  1 is exactly divisible by what two numbers
between 60 and 70?
 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?
 Let A be nonzero square matrix with the property that A^{3} = 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 B^{2} 0, B^{3} = 0.
 Define F(x) by
F(x) = S_{n = 0}^{}F_{n}x^{n} (wherever the
series converges), where F_{n} is the nth Fibonacci number defined
by
F_{0} = F_{1} = 1,
F_{n} = F_{n  1} + F_{n  2}, n > 1. Find an
explicit closed form for F(x).
 Two elements A, B in a group G have the property
ABA^{1}B = 1,
where 1 denotes the identity element in G.
 (a)
 Show that
AB^{2} = B^{2}A.
 (b)
 Show that
AB^{n} = B^{n}A for any integer n.
 (c)
 Find u and v so that
(B^{a}A^{b})(B^{c}A^{d}) = B^{u}A^{v}.
 With k a positive integer, prove that
(1  k^{2})^{k}1  1/k.
 Let
A = {a_{0}, a_{1},...} be a sequence of real numbers and
define the sequence
A' = {a_{0}', a_{1}',...} as follows for
n = 0, 1,...:
a_{2n}' = a_{n},
a_{2n + 1}' = a_{n} + 1. If a_{0} = 1 and
A' = A, find
 (a)

a_{1}, a_{2}, a_{3} and a_{4}
 (b)
 a_{1981}
 (c)
 A simple general algorithm for evaluating a_{n}, for
n = 0, 1,....
 Let
 (i)
 0 < a < 1,
 (ii)

0 < M_{k + 1} < M_{k}, for
k = 0, 1,...,
 (iii)

lim_{k  > }M_{k} = 0.
If
b_{n} = S_{k = 0}^{}a^{n  k}M_{k}, prove that
lim_{n  > }b_{n} = 0.
Peter Linnell
20010812