24th Annual
Virginia Tech Regional Mathematics Contest
From 8:30a.m. to 11:00a.m., October 26, 2002

Fill out the individual registration form

1. Let a, b be positive constants. Find the volume (in the first octant) which lies above the region in the xy-plane bounded by x = 0, x = p/2, y = 0, y$\sqrt{b^2 \cos^2 x + a^2 \sin^2 x}$ = 1, and below the plane z = y.

2. Find rational numbers a, b, c, d, e such that
   $$\sqrt{7 + \sqrt{40}}$$ = a + b$$\sqrt{2}$$ + c$$\sqrt{5}$$ + d$$\sqrt{7}$$ + e$$\sqrt{10}$$.

3. Let A and B be nonempty subsets of S = {1, 2,..., 99} (integers from 1 to 99 inclusive). Let a and b denote the number of elements in A and B respectively, and suppose a + b = 100. Prove that for each integer s in S, there are integers x in A and y in B such that x + y = s or s + 99.

4. Let {1,2,3,4} be a set of abstract symbols on which the associative binary operation * is defined by the following operation table (associative means (a*b)*c = a*(b*c)):

   *	1	2	3	4
   1	1	2	3	4
   2	2	1	4	3
   3	3	4	1	2
   4	4	3	2	1

   If the operation * is represented by juxtaposition, e.g., 2*3 is written as 23 etc., then it is easy to see from the table that of the four possible ``words" of length two that can be formed using only 2 and 3, i.e., 22, 23, 32 and 33, exactly two, 22 and 33, are equal to 1. Find a formula for the number A(n) of words of length n, formed by using only 2 and 3, that equal 1. From the table and the example just given for words of length two, it is clear that A(1) = 0 and A(2) = 2. Use the formula to find A(12).

5. Let n be a positive integer. A bit string of length n is a sequence of n numbers consisting of 0's and 1's. Let f (n) denote the number of bit strings of length n in which every 0 is surrounded by 1's. (Thus for n = 5, 11101 is allowed, but 10011 and 10110 are not allowed, and we have f (3) = 2, f (4) = 3.) Prove that f (n) < (1.7)ⁿ for all n.

6. Let S be a set of 2 X 2 matrices with complex numbers as entries, and let T be the subset of S consisting of matrices whose eigenvalues are ±1 (so the eigenvalues for each matrix in T are {1, 1} or {1, - 1} or { - 1, - 1}). Suppose there are exactly three matrices in T. Prove that there are matrices A, B in S such that AB is not a matrix in S (A = B is allowed).

7. Let {a_n}_n>1 be an infinite sequence with a_n> 0 for all n. For n>1, let b_n denote the geometric mean of a₁,..., a_n, that is (a₁...a_n)^1/n. Suppose S_{n = 1}^$\infty$a_n is convergent. Prove that S_{n = 1}^$\infty$b_n² is also convergent.