3rd Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 7, 1981

Fill out the individual registration form

1. The number 2⁴⁸ - 1 is exactly divisible by what two numbers between 60 and 70?

2. For which real numbers b does the function f (x), defined by the conditions f (0) = b and f' = 2f - x, satisfy f (x) > 0 for all x$\ge$ 0?

3. Let A be non-zero square matrix with the property that A³ = 0, where 0 is the zero matrix, but with A being otherwise arbitrary.

   (a)

   Express (I - A)^-1 as a polynomial in A, where I is the identity matrix.

   (b)

   Find a 3 X 3 matrix satisfying B²$\ne$ 0, B³ = 0.

4. Define F(x) by F(x) = S_{n = 0}^$\infty$F_nxⁿ (wherever the series converges), where F_n is the nth Fibonacci number defined by F₀ = F₁ = 1, F_n = F_{n - 1} + F_{n - 2}, n > 1. Find an explicit closed form for F(x).

5. Two elements A, B in a group G have the property ABA^-1B = 1, where 1 denotes the identity element in G.

   (a)

   Show that AB² = B^-2A.

   (b)

   Show that ABⁿ = B^-nA for any integer n.

   (c)

   Find u and v so that (B^aA^b)(B^cA^d) = B^uA^v.

6. With k a positive integer, prove that (1 - k^-2)^k$\ge$1 - 1/k.

7. Let A = {a₀, a₁,...} be a sequence of real numbers and define the sequence A' = {a₀', a₁',...} as follows for n = 0, 1,...: a_2n' = a_n, a_{2n + 1}' = a_n + 1. If a₀ = 1 and A' = A, find

   (a)

   a₁, a₂, a₃ and a₄

   (b)

   a₁₉₈₁

   (c)

   A simple general algorithm for evaluating a_n, for n = 0, 1,....

8. Let

   (i)

   0 < a < 1,

   (ii)

   0 < M_{k + 1} < M_k, for k = 0, 1,...,

   (iii)

   lim_{k - > $\infty$}M_k = 0.

   If b_n = S_{k = 0}^$\infty$a^{n - k}M_k, prove that lim_{n - > $\infty$}b_n = 0.