Virginia Tech Regional Math Contest (1999)

Problem 1

Let $G$ be the set of all continuous functions $f : \mathbb{R} \to \mathbb{R}$, satisfying the following properties:

Show that there is a number $\alpha$ such that $\alpha = \displaystyle{\int_{0}^{1} \int_{0}^{x} f(x + y) dy dx}$ for all $f \in G$.

Problem 2

Suppose that $f : \mathbb{R} \to \mathbb{R}$ is infinitely differentiable and satisfies both of the following properties:

Find $f(x)$. Guesswork will not be accepted.

Problem 3

Let $\varepsilon, M$ be positive real numbers, and let $A_1, A_2, \dots$ be a sequence of matrices such that for all $n$,

If $\delta$ is a positive real number, let $e_n(\delta)$ denote the number of nonzero eigenvalues of $A_n$ which have absolute value less that $\delta$. (Some eigenvalues can be complex numbers.) Prove that one can choose $\delta > 0$ so that $e_n(\delta)/n < \varepsilon$ for all $n$.

Problem 4

A rectangular box has sides of length 3, 4, 5. Find the volume of the region consisting of all points that are within distance 1 of at least one of the sides.

Problem 5

Let $f : \mathbb{R}_+ \to \mathbb{R}_+$ be a function from the set of positive real numbers to the same set satisfying $f(f(x)) = x$ for all positive $x$. Suppose that $f$ is infinitely differentiable for all positive $x$, and that $f(a) \ne a$ for some positive $a$. Prove that $\displaystyle{\lim_{x \to \infty} f(x) = 0}$.

Problem 6

A set $S$ of distinct positive integers has property ND if no element $x$ of $S$ divides the sum of the integers in any subset of $S \setminus {x}$. Here $S \setminus {x}$ means the set that remains after $x$ is removed from $S$.