16th Annual
Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 29, 1994

Fill out the individual registration form

1. Evaluate e(1 - z)2 dzdydx.

2. 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 0xa, 0yf (a), we have

xyf (t) dt + g(t) dt.

3. Find all continuously differentiable solutions f (x) for

f (x)2 = (f (t)2 - f (t)4 + (f'(t))2) dt + 100

where f (0)2 = 100.

4. Consider the polynomial equation ax4 + bx3 + x2 + 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.

5. 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

for all 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).

6. Let A be an n X n matrix and let a be an n-dimensional vector such that Aa = 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 ai/|bi|, i = 1,..., n, where the ai's and bi's are the components of a and b, respectively.

7. Define f (1) = 1 and f (n + 1) = 2 for n1. If N1 is an integer, find Sn = 1Nf (n)2.

8. Let a sequence {xn}n = 0 of rational numbers be defined by x0 = 10, x1 = 29 and xn + 2 = 19xn + 1/(94xn) for n 0. Find Sn = 0x6n/2n.

Peter Linnell
2000-09-08