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,....

    Find limn - > $\scriptstyle \infty$ and give a proof of your answer.

  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 $ \int_{0}^{1}$tnp(t) dt = n for n = 0, 1, 2.

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

    limr - > 0+($\displaystyle \int_{0}^{1}$(f (x))r dx)1/r = e$\scriptstyle \int_{0}^{1}$ln 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 $ \in$ B there exists a $ \in$ A such that f (a) = b.)

  7. A function f from the positive integers to the positive integers has the properties:

    (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