8th Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 1, 1986
Fill out the individual registration form
- Let x1 = 1, x2 = 3, and
xn + 1 = (1/(n + 1))Si = 1nxi forn = 2, 3,....
Find
limn - > and give a proof of your answer.
- Given that a > 0 and c > 0, find a necessary and sufficient condition
on b so that
ax2 + bx + c > 0 for all x > 0.
- Express sinh 3x as a polynomial in sinh x. As an example, the
identity
cos 2x = 2cos2x - 1 shows that cos 2x can be
expressed as a polynomial in cos x. (Recall that sinh denotes the
hyperbolic sine defined by
sinhx = (ex - e-x)/2.)
- Find the quadratic polynomial
p(t) = a0 + a1t + a2t2 such that
tnp(t) dt = n for n = 0, 1, 2.
- Verify that, for
f (x) = x + 1,
lim
r - > 0+(
(
f (
x))
r d
x)
1/r =
eln f(x) dx.
- Sets A and B are defined by
A = {1, 2,..., n} and
B = {1, 2, 3}. Determine the number of distinct functions from A onto
B. (A function
f : A - > B is ``onto" if for each b B
there exists a A such that f (a) = b.)
- A function f from the positive integers to the positive integers
has the properties:
- f (1) = 1,
- f (n) = 2 if n100,
-
f (n) = f (n/2) if n is even and n < 100,
-
f (n) = f (n2 + 7) if n is odd and n > 1.
- (a)
- Find all positive integers n for which the stated properties
require that f (n) = 1.
- (b)
- Find all positive integers n for which the stated properties do not
determine f (n).
- Find all pairs N, M of positive integers, N < M, such that
Sj=NM 1/(j(j + 1)) = 1/10.
Peter Linnell
2001-08-12