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
   x_{n + 1} = (1/(n + 1))S_{i = 1}ⁿx_i    forn = 2, 3,....
   Find lim_{n - > $\infty$} 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 = (e^x - 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,
   lim_{r - > 0⁺}($$\int_{0}^{1}$$(f (x))^r dx)^1/r = e^{$\int_{0}^{1}$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:
   • f (1) = 1,

   • f (n) = 2 if n$\ge$100,

   • 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
   S_j=N^M 1/(j(j + 1)) = 1/10.