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

    $ \int_{0}^{1}$f (x) dx = 1999.
    Show that there is a number a such that a = $\displaystyle \int_{0}^{1}$$\displaystyle \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.
    f (1) = 2,

    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,
    An is an n X n matrix with integer entries,

    The sum of the absolute values of the entries in each row of An is at most M.
    If d is a positive real number, let en(d) denote the number of nonzero eigenvalues of An which have absolute value less that d. (Some eigenvalues can be complex numbers.) Prove that one can choose d > 0 so that en(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 $\displaystyle \lim_{x -> \infty}^{}$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}$.
    Find the smallest positive integer n such that {3, 4, n} has property ND.

    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.

Peter Linnell