21st Annual
Virginia Tech Regional Mathematics Contest
From 8:30a.m. to 11:00a.m., October 30, 1999

Fill out the individual registration form

1. Let $\mathcal {G}$ be the set of all continuous functions f : $\mathbb {R}$ - > $\mathbb {R}$, satisfying the following properties.

   (i)

   f (x) = f (x + 1) for all x,

   (ii)

   $\int_{0}^{1}$f (x) dx = 1999.

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

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

   (i)

   f (1) = 2,

   (ii)

   If a,b are real numbers satisfying a² + b² = 1, then f (ax)f (bx) = f (x) for all x.

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

3. Let e, M be positive real numbers, and let A₁, A₂,... be a sequence of matrices such that for all n,

   (i)

   A_n is an n X n matrix with integer entries,

   (ii)

   The sum of the absolute values of the entries in each row of A_n is at most M.

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

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.

5. Let f : $\mathbb {R}$₊ - > $\mathbb {R}$₊ 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 $$\lim_{x -> \infty}^{}$$f (x) = 0.

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

   (i)

   Find the smallest positive integer n such that {3, 4, n} has property ND.

   (ii)

   If n is the number found in (i), prove that no set $\mathcal {S}$ with property ND has {3, 4, n} as a proper subset.