Problem 1
Let $G$ be the set of all continuous functions $f : \mathbb{R} \to \mathbb{R}$, satisfying the following properties:
- $f(x) = f(x + 1)$ for all $x$,
- $\displaystyle{\int_{0}^{1} f(x) dx = 1999}$.
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:
- $f(1) = 2$,
- If $\alpha, \beta$ are real numbers satisfying $\alpha^2 + \beta^2 = 1$, then $f(\alpha x) f(\beta x) = f(x)$ for all $x$.
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 $n$,
- $A_n$ is an $n \times n$ matrix with integer entries,
- The sum of the absolute values of the entries in each row of $A_n$ is at most $M$.
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 $S$.
- Find the smallest positive integer $n$ such that $\lbrace 3, 4, n \rbrace$ has property ND.
- If $n$ is the number found in (i), prove that no set $S$ with property ND has $\lbrace 3, 4, n \rbrace$ as a proper subset.