21st Annual
Virginia Tech Regional Mathematics Contest
From 8:30a.m. to 11:00a.m., October 30, 1999
Fill out the individual registration form
 Let
be the set of all continuous functions
f :  > , satisfying the following properties.
 (i)

f (x) = f (x + 1) for all x,
 (ii)

f (x) dx = 1999.
Show that there is a number
a such that
a = f (x + y) dydx
for all
f .
 Suppose that
f :  > is infinitely
differentiable and satisfies both of the following properties.
 (i)
 f (1) = 2,
 (ii)
 If
a,b are real numbers satisfying
a^{2} + b^{2} = 1, then
f (ax)f (bx) = f (x) for all x.
Find f (x). Guesswork will not be accepted.
 Let
e, M be positive real numbers, and let
A_{1}, A_{2},...
be a sequence of matrices such that for all n,
 (i)
 A_{n} is an n X n matrix with integer entries,
 (ii)
 The sum of the absolute values of the entries in each row of
A_{n} is at most M.
If
d is a positive real number, let
e_{n}(d) denote the
number of nonzero eigenvalues of A_{n} which have absolute value less
that
d. (Some eigenvalues can be complex numbers.) Prove
that one can choose
d > 0 so that
e_{n}(d)/n < e
for all n.
 A rectangular box has sides of length 3, 4, 5. Find the volume of
the region consisting of all points that are within distance 1 of at
least one of the sides.
 Let
f : _{+}  > _{+} be a function from the
set of positive real numbers to the same set satisfying
f (f (x)) = x
for all positive x. Suppose that f is infinitely differentiable
for all positive X, and that
f (a)a for some positive a.
Prove that
f (x) = 0.
 A set
of distinct positive integers has property
ND if no element x of
divides the sum of
the integers in any subset of
S\{x}. Here
\{x} means the set that remains after x is removed
from
.
 (i)
 Find the smallest positive integer n such that {3, 4, n} has
property ND.
 (ii)
 If n is the number found in (i), prove that no set
with property ND has {3, 4, n} as a proper
subset.
Peter Linnell
20000908