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 $$\int_{0}^{1}$$$$\int_{0}^{x}$$$$\int_{0}^{1-x^2}$$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 0$\le$x$\le$a, 0$\le$y$\le$f (a), we have
   xy$$\le$$$$\int_{0}^{x}$$f (t) dt + $$\int_{0}^{y}$$g(t) dt.

3. Find all continuously differentiable solutions f (x) for
   f (x)² = $$\int_{0}^{x}$$(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 : $\mathbb {Z}$ X $\mathbb {Z}$ - > $\mathbb {R}$ 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 $\in$ $\mathbb {Z}$ (where $\mathbb {Z}$ and $\mathbb {R}$ denote the set of all integers and all real numbers, respectively). Prove that | f (m, n)|$\ge$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 a_i/|b_i|, i = 1,..., n, where the a_i's and b_i's are the components of a and b, respectively.

7. Define f (1) = 1 and f (n + 1) = 2$\sqrt{f(n)^2 + n}$ for n$\ge$1. If N$\ge$1 is an integer, find S_{n = 1}^Nf (n)².

8. Let a sequence {x_n}_{n = 0}^$\infty$ of rational numbers be defined by x₀ = 10, x₁ = 29 and x_{n + 2} = 19x_{n + 1}/(94x_n) for n$\ge$ 0. Find S_{n = 0}^$\infty$x_6n/2ⁿ.