8th Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 1, 1986

Fill out the individual registration form

1. Let x1 = 1, x2 = 3, and

xn + 1 = (1/(n + 1))Si = 1nxi    forn = 2, 3,....

2. 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.

3. 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.)

4. Find the quadratic polynomial p(t) = a0 + a1t + a2t2 such that tnp(t) dt = n for n = 0, 1, 2.

5. Verify that, for f (x) = x + 1,

limr - > 0+((f (x))r dx)1/r = eln f(x) dx.

6. 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.)

7. 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).

8. 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