Virginia Tech Regional Mathematics Contest

- Let
*I*denote the 2 X 2 identity matrix( 1 0 0 1 ) *M*=( *I**A**B**C*), *N*=( *I**B**A**C*) where

*A*,*B*,*C*are arbitrary 2 X 2 matrices which entries in**R**, the real numbers. Thus*M*and*N*are 4 X 4 matrices with entries in**R**. Is it true that*M*is invertible (i.e. there is a 4 X 4 matrix*X*such that*MX*=*XM*= the identity matrix) implies*N*is invertible? Justify your answer. - A sequence of integers {
*f*(*n*)} for*n*= 0, 1, 2,... is defined as follows:*f*(0) = 0 and for*n*> 0,*f*(*n*) =*f*(*n*- 1) + 3, if*n*= 0 or 1 (mod 6),*f*(*n*- 1) + 1, if*n*= 2 or 5 (mod 6),*f*(*n*- 1) + 2, if*n*= 3 or 4 (mod 6).

Derive an explicit formula for

*f*(*n*) when*n*= 0 (mod 6), showing all necessary details in your derivation. - A computer is programmed to randomly generate a string of six symbols
using only the letters
*A*,*B*,*C*. What is the probability that the string will not contain three consecutive*A*'s? - A
9 X 9 chess board has two squares from opposite corners and
its central square removed (so 3 squares on the same diagonal are
removed, leaving 78 squares). Is it possible to cover the remaining
squares using dominoes, where each domino covers two adjacent
squares? Justify your answer.
- Let
*f*(*x*) = sin(*t*^{2}-*t*+*x*) d*t*. Compute*f''*(*x*) +*f*(*x*) and deduce that*f*^{(12)}(0) +*f*^{(10)}(0) = 0 (*f*^{(10)}indicates 10th derivative).

(Please turn over) - An enormous party has an infinite number of people. Each two
people either know or don't know each other. Given a
positive integer
*n*, prove there are*n*people in the party such that either they all know each other, or nobody knows each other (so the first possibility means that if*A*and*B*are any two of the*n*people, then*A*knows*B*, whereas the second possibility means that if*A*and*B*are any two of the*n*people, then*A*does not know*B*). - Let {
*a*_{n}} be a sequence of positive real numbers such that lim_{n--> }*a*_{n}= 0. Prove that S_{n=1}^{}| 1 -*a*_{n+1}/*a*_{n}| is divergent.

Peter Linnell 2004-10-29