Virginia Tech Regional Mathematics Contest

- Evaluate
*e*^{(1 - z)2}d*zdydx*. - Let
*f*be continuous real function, strictly increasing in an interval [0,*a*] such that*f*(0) = 0. Let*g*be the inverse of*f*, i.e.,*g*(*f*(*x*)) =*x*for all*x*in [0,*a*]. Show that for 0*x**a*, 0*y**f*(*a*), we have*xy**f*(*t*) d*t*+*g*(*t*) d*t*. - Find all continuously differentiable solutions
*f*(*x*) for*f*(*x*)^{2}= (*f*(*t*)^{2}-*f*(*t*)^{4}+ (*f'*(*t*))^{2}) d*t*+ 100*f*(0)^{2}= 100. - Consider the polynomial equation
*ax*^{4}+*bx*^{3}+*x*^{2}+*bx*+*a*= 0, where*a*and*b*are real numbers, and*a*> 1/2. Find the maximum possible value of*a*+*b*for which there is at least one positive real root of the above equation. - Let
*f*: X - > be a function which satisfies*f*(0, 0) = 1 and*f*(*m*,*n*) +*f*(*m*+ 1,*n*) +*f*(*m*,*n*+ 1) +*f*(*m*+ 1,*n*+ 1) = 0*m*,*n*(where and denote the set of all integers and all real numbers, respectively). Prove that |*f*(*m*,*n*)|1/3, for infinitely many pairs of integers (*m*,*n*). - Let
*A*be an*n*X*n*matrix and let a be an*n*-dimensional vector such that*A*a = a. Suppose that all the entries of*A*and a are positive real numbers. Prove that a is the only linearly independent eigenvector of*A*corresponding to the eigenvalue 1. Hint: if b is another eigenvector, consider the minimum of a_{i}/|b_{i}|,*i*= 1,...,*n*, where the a_{i}'s and b_{i}'s are the components of a and b, respectively. - Define
*f*(1) = 1 and*f*(*n*+ 1) = 2 for*n*1. If*N*1 is an integer, find S_{n = 1}^{N}*f*(*n*)^{2}. - Let a sequence
{
*x*_{n}}_{n = 0}^{}of rational numbers be defined by*x*_{0}= 10,*x*_{1}= 29 and*x*_{n + 2}= 19*x*_{n + 1}/(94*x*_{n}) for*n*0. Find S_{n = 0}^{}*x*_{6n}/2^{n}.