Virginia Tech Regional Mathematics Contest

- 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*= 1, and below the plane*z*=*y*. - Find rational numbers
*a*,*b*,*c*,*d*,*e*such that=*a*+*b*+*c*+*d*+*e*. - 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. - 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). - 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*. - 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). - 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*a*_{1},...,*a*_{n}, that is (*a*_{1}...*a*_{n})^{1/n}. Suppose S_{n = 1}^{}*a*_{n}is convergent. Prove that S_{n = 1}^{}*b*_{n}^{2}is also convergent.